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This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur as such Galois groups. Also the explicit structure of ray class groups of regular cyclotomic number field is given.

Let $q = p^s$ be a power of a prime number $p$ and let $\mathbb{F}_q$ be the finite field with $q$ elements. In this paper we obtain the explicit factorization of the cyclotomic polynomial $\Phi_{2^nr}$ over $\mathbb{F}_q$ where both $r \geq 3$ and $q$ are odd, $\gcd(q,r) = 1$, and $n\in \mathbb{N}$. Previously, only the special cases when $r = 1,\ 3,\ 5$ had been achieved. For this we make the assumption that the explicit factorization of $\Phi_r$ over $\mathbb{F}_q$ is given to us as a known. Let $n = p_1^{e_1}p_2^{e_2}... p_s^{e_s}$ be the factorization of $n \in \mathbb{N}$ into powers of distinct primes $p_i,\ 1\leq i \leq s$. In the case that the orders of $q$ modulo all these prime powers $p_i^{e_i}$ are pairwise coprime we show how to obtain the explicit factors of $\Phi_{n}$ from the factors of each $\Phi_{p_i^{e_i}}$. We also demonstrate how to obtain the factorization of $\Phi_{mn}$ from the factorization of $\Phi_n$ when $q$ is a primitive root modulo $m$ and $\gcd(m,n) = \gcd(\phi(m),\ord_n(q)) = 1$. Here $\phi$ is the Euler's totient function, and $\ord_n(q)$ denotes the multiplicative order of $q$ modulo $n$. Moreover, we present the construction of a new class of irreducible polynomials over $\mathbb{F}_q$ and generalize a result due to Varshamov (1984) \cite{Varshamov}.

This article deals with a study of the structure of the class group of the cyclotomic field $K=\Q(\zeta_p)$ for $p$ an odd prime number, starting from Stickelberger relation. The present state of this work leads me to set a question for all the prime numbers $h\not=p$ which divide the relative class number $h^-(K)$.

Let $\ell$ and $p$ be odd primes. For a positive integer $\mu$ let $k_\mu$ be the ray class field of $k=\mathbb{Q}(e^{2\pi i/\ell})$ modulo $2p^\mu$. We present certain class fields $K_\mu$ of $k$ such that $k_\mu\leq K_\mu\leq k_{\mu+1}$, and find the degree of $K_\mu/k_\mu$ explicitly. And we also construct, in the sense of Hilbert, primitive generators of the field $K_\mu$ over $k_\mu$ by using Shimura's reciprocity law and special values of theta constants.

In this paper we study the possible torsions of elliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt {-3})$.

Let $\zeta_k$ be a $k$-th primitive root of unity, $m\geq\phi(k)+1$ an integer and $\Phi_k(X)\in\mathbb Z [X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+\zeta_k,\mathcal N)$ is a canonical number system, with $\mathcal N=\{0,1,\dots,|\Phi_k(m)|\}$. Moreover we also discuss whether the two bases $-m+\zeta_k$ and $-n+\zeta_k$ are multiplicatively independent for positive integers $m$ and $n$ and $k$ fixed.

In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation polynomials that are involutions. Our results provide a fast algorithm (only modular operations are involved) to generate many classes of generalized cyclotomic permutation polynomials, their inverses, and involutions.

We use a result of Thaine to give an alternative proof of the fact that, for a prime p>3 congruent to 3 modulo 4, the component e_{(p+1)/2} of the p-Sylow subgroup of the ideal class group of \mathbb Q(\zeta_{p}) is trivial.

We study the explicit factorization of $2^n r$-th cyclotomic polynomials over finite field $\mathbb{F}_q$ where $q, r$ are odd with $(r, q) =1$. We show that all irreducible factors of $2^n r$-th cyclotomic polynomials can be obtained easily from irreducible factors of cyclotomic polynomials of small orders. In particular, we obtain the explicit factorization of $2^n 5$-th cyclotomic polynomials over finite fields and construct several classes of irreducible polynomials of degree $2^{n-2}$ with fewer than 5 terms. The reciprocals of these irreducible polynomials are irreducible polynomials of the form $x^{2^{n-2}} + g(x)$ such that the degree of $g(x)$ is small ($\leq 4$), which could have potential applications as mentioned by Gao, Howell, and Panario in \cite{GaoHowellPanario}.

We first study some families of maximal real subfields of cyclotomic fields with even class number, and then explore the implications of large plus class numbers of cyclotomic fields. We also discuss capitulation of the minus part and the behaviour of p-class groups in cyclic ramified p-extensions.

We study two criterions of cyclicity for divisor class groups of functions fields, the first one involves Artin L-functions and the second one involves "affine" class groups. We show that, in general, these two criterions are not linked.

In this note, we investigate the p-adic behavior of Weil numbers in the cyclotomic $\mathbb Z\_p$-extension of $\mathbb Q(\zeta\_p).$ We determlne the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in Iwasawa Theory.

Let p be an odd prime. Let K = \Q(zeta) be the p-cyclotomic number field. Let v be a primitive root mod p and sigma : zeta --> zeta^v be a \Q-isomorphism of the extension K/\Q generating the Galois group G of K/\Q. For n in Z, the notation v^n is understood by v^n mod p with 1 \leq v^n \leq p-1. Let P(X) = \sum_{i=0}^{p-2} v^{-i}X^i \in \Z[X] be the Stickelberger polynomial. P(sigma) annihilates the class group C of K. There exists a polynomial Q(X) \in \Z[X] such that P(sigma)(sigma-v) = p\times Q(sigma) and such that Q(sigma) annihilates the p-class group C_p of K (the subgroup of exponent p of C). In the other hand sigma^{(p-1)/2}+1 annihilates the relative class group of K. The simultaneous application of these results brings some informations on the structure of the class group C, give some explicit congruences in \Z[v] mod p for the p-class group C_p of K and some explicit congruences in \Z[v] mod h for the h-class group of K for all the prime divisors h \not = p of the class number h(K). We detail at the end the case of class number of quadratic and biquadratic fields contained in the cyclotomic field K and give a general MAPLE algorithm.

Let $p$ be an irregular prime and $K=\Q(\zeta)$ the $p$-cyclotomic field. Let $\sigma$ be a $\Q$-isomorphism of $K$ generating $Gal(K/\Q)$. Let $S/K$ be a cyclic unramified extension of degree $p$, defined by $S= K(A^{1/p})$ where $A\in K\backslash K^p$, $A\Z_K=\mk a^p$ with $\mk a$ non-principal ideal of $\Z_K$, $A^{\sigma-\mu}\in K^p$ and $\mu\in{\bf F}_p$. We compute explicitly the decomposition of the prime $p$ in the subfields $M$ of $S$ of degree $[M:\Q]=p$.

Recently the authors showed that the algebraic integers of the form $-m+\zeta_k$ are bases of a canonical number system of $\mathbb{Z}[\zeta_k]$ provided $m\geq \phi(k)+1$, where $\zeta_k$ denotes a $k$-th primitive root of unity and $\phi$ is Euler's totient function. In this paper we are interested in the questions whether two bases $-m+\zeta_k$ and $-n+\zeta_k$ are multiplicatively independent. We show the multiplicative independence in case that $0

In this article we prove that (1-zeta+zeta^2) is a unit in the ring of integers of the cyclotomic field where zeta is a primitive n-th root of unity and n is coprime to 2 and 3. We also prove that for prime n, N_{Q(zeta)/Q}(1-zeta-zeta^2)=L(p) the p-th Lucas number thus completing the study of norms of quadratic polynomials in zeta that only have coefficients equal to 1 or -1 and both numbers appear.

Algebraic codes that achieve list decoding capacity were recently constructed by a careful ``folding'' of the Reed-Solomon code. The ``low-degree'' nature of this folding operation was crucial to the list decoding algorithm. We show how such folding schemes conducive to list decoding arise out of the Artin-Frobenius automorphism at primes in Galois extensions. Using this approach, we construct new folded algebraic-geometric codes for list decoding based on cyclotomic function fields with a cyclic Galois group. Such function fields are obtained by adjoining torsion points of the Carlitz action of an irreducible $M \in \F_q[T]$. The Reed-Solomon case corresponds to the simplest such extension (corresponding to the case $M=T$). In the general case, we need to descend to the fixed field of a suitable Galois subgroup in order to ensure the existence of many degree one places that can be used for encoding. Our methods shed new light on algebraic codes and their list decoding, and lead to new codes achieving list decoding capacity. Quantitatively, these codes provide list decoding (and list recovery/soft decoding) guarantees similar to folded Reed-Solomon codes but with an alphabet size that is only polylogarithmic in the block length. In comparison, for folded RS codes, the alphabet size is a large polynomial in the block length. This has applications to fully explicit (with no brute-force search) binary concatenated codes for list decoding up to the Zyablov radius.

Algebraic codes that achieve list decoding capacity were recently constructed by a careful ``folding'' of the Reed-Solomon code. The ``low-degree'' nature of this folding operation was crucial to the list decoding algorithm. We show how such folding schemes conducive to list decoding arise out of the Artin-Frobenius automorphism at primes in Galois extensions. Using this approach, we construct new folded algebraic-geometric codes for list decoding based on cyclotomic function fields with a cyclic Galois group. Such function fields are obtained by adjoining torsion points of the Carlitz action of an irreducible $M \in \F_q[T]$. The Reed-Solomon case corresponds to the simplest such extension (corresponding to the case $M=T$). In the general case, we need to descend to the fixed field of a suitable Galois subgroup in order to ensure the existence of many degree one places that can be used for encoding. Our methods shed new light on algebraic codes and their list decoding, and lead to new codes achieving list decoding capacity. Quantitatively, these codes provide list decoding (and list recovery/soft decoding) guarantees similar to folded Reed-Solomon codes but with an alphabet size that is only polylogarithmic in the block length. In comparison, for folded RS codes, the alphabet size is a large polynomial in the block length. This has applications to fully explicit (with no brute-force search) binary concatenated codes for list decoding up to the Zyablov radius.

We prove that $\mu = 0$ for the cyclotomic $\Z_p$-extensions of CM number fields.

A representation of finite fields that has proved useful when implementing finite field arithmetic in hardware is based on an isomorphism between subrings and fields. In this paper, we present an unified formulation for multiplication in cyclotomic rings and cyclotomic fields in that most arithmetic operations are done on vectors. From this formulation we can generate optimized algorithms for multiplication. For example, one of the proposed algorithms requires approximately half the number of coordinate-level multiplications at the expense of extra coordinate-level additions. Our method is then applied to the finite fields GF(q^m) to further reduce the number of operations. We then present optimized algorithms for multiplication in finite fields with type-I and type-II optimal normal bases.

Hilbert's Satz 90 tells us that for a given cyclic extension $K/k$, a unit of norm $1$ in $K$ can be written as a quotient of conjugate elements in $K$. For the extensions $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ with $p$ prime $> 3$, Newman proved a refinement of Hilbert's Satz 90 that gives a sufficient and necessary condition for which a unit of norm $1$ in $\mathbb{Q}(\zeta_p)$ can be written as a quotient of conjugate units. In order to obtain this result, Newman proved a stronger result that gives a unique representation of units of norm $1$ as a product of a power of $\dfrac{1 - \zeta_p^e}{1 - \zeta_p}$ with a quotient of conjugate units, where $e$ is a given primitive root modulo $p$. In this paper, we obtain a function field analogue of Newman's result for the $\wp$-th cyclotomic function field extensions $\mathbb{K}_{\wp}/\mathbb{F}_q(T)$, where $\wp$ is a monic prime in $\mathbb{F}_q[T]$. As a consequence, we proved a refinement of Hilbert's Satz 90 for the extensions $\mathbb{K}_{\wp}/\mathbb{F}_q(T)$ that gives a sufficient and necessary condition for which a unit of norm $1$ in $\mathbb{K}_{\wp}$ can be written as a quotient of conjugate units.

In this essay, we see how prime cyclotomic fields (cyclotomic fields obtained by adjoining a primitive p-th root of unity to Q, where p is an odd prime) can lead to elegant proofs of number theoretical concepts. We namely develop the notion of primary units in a cyclotomic field, demonstrate their equivalence to real units in this case, and show how this leads to a proof of a special case of Fermat's Last Theorem. We finally modernize Dirichlet's solution to Pell's Equation.

In this essay, we see how prime cyclotomic fields (cyclotomic fields obtained by adjoining a primitive p-th root of unity to Q, where p is an odd prime) can lead to elegant proofs of number theoretical concepts. We namely develop the notion of primary units in a cyclotomic field, demonstrate their equivalence to real units in this case, and show how this leads to a proof of a special case of Fermat's Last Theorem. We finally modernize Dirichlet's solution to Pell's Equation.

Let $p$ be an irregular prime. Let $K=\Q(\zeta)$ be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$. We give an algebraic construction of the subfields $M$ of $S$ with degree $[M:\Q]=p$ and an explicit formula for the prime decomposition and ramification of the prime number $p$ in the extensions $S/K$, $M/\Q$ and $S/M$. In the last section, we examine the consequences of these results for the Vandiver's conjecture. This article is at elementary level on Classical Algebraic Number Theory.

Let p > 2 be a prime. Let Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of Q(zeta) lying over p. This article aims to describe some pi-adic congruences characterizing the structure of the p-class group and of the unit group of the field Q(zeta). For the unit group, this article supplements the 1954 and 1956 papers of Denes on this topic. A complete summarizing of the results obtained follows in the Introduction section of the paper (pages 3 to 6). This new version of the article with the same title, submitted with reference math.NT/0407430 25 Jul 2004: - corrects several typing errors in the introduction and in the paper, - simplifies some proofs of pi-adic congruences connected to p-class group, - removes the section dealing of the singular group foreseen in an independant paper.

In 1977 Kervaire and Murthy presented three conjectures regarding $K_0 \mathbb{Z} C_{p^n}$, where $C_{p^n}$ is the cyclic group of order $p^n$ and $p$ is a semi-regular prime. The Mayer-Vietoris exact sequence provides the following short exact sequence $$0\to V_n\to \pic (\mathbb{Z} C_{p^n})\to \cl \mathbb{Q} (\zeta_{n-1})\times \pic (\mathbb{Z} C_{p^{n-1}})\to 0$$ where $\zeta_{n-1}$ is a primitive $p^n$-th root of unity. The group $V_n$ that injects into $\pic \mathbb{Z} C_{p^n}\cong\tilde{K}_0\mathbb{Z} C_{p^n}$, is a canonical quotient of an in some sense simpler group $\mathcal{V}_n$. Both groups split in a ``positive'' and ``negative'' part. While $V_n^-$ is well understood there is still no complete information on $V_n^+$. Kervaire and Murthy showed that $K_0 \Z C_{p^n}$ and $V_n$ are tightly connected to class groups of cyclotomic fields. They conjectured that $V_n^+\cong (\mathbb{Z}/p^n \mathbb{Z})^{r(p)}$, where $r(p)$ is the index of regularity of the prime $p$ and that $\mathcal{V}_n^+\cong V_n^+$, and moreover, $\char\mathcal{V}_n^+\cong \cl^{(p)} \mathbb{Q} (\zeta_{n-1})$, the $p$-part of the class group. In the present paper we calculate $\mathcal{V}_n^+$ and prove that $\char \mathcal{V}_n^+\cong \cl^{(p)} \mathbb{Q}(\zeta_{n-1})$ for all semi-regular primes which also gives us the structure of $\cl^{(p)} \mathbb{Q}(\z_{n-1})$ as an abelian group. Moreover we conclude that all three Kervaire and Murthy conjectures hold is equivalent to that the Iwasawa invariant $\lambda$ equals $r(p)$ and that this also implies that the Iwasawa invariant $\nu$ equals $r(p)$.

Let $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(\sqrt{-1})$ and let $C_n$ denote the cyclic group of order $n$. We study how the torsion part of an elliptic curve over $K$ grows in a quadratic extension of $K$. In the case $E(K)[2] \approx C_2 \oplus C_2$ we determine how a given torsion structure can grow in a quadratic extension and the maximum number of quadratic extensions in which it grows. We also classify the torsion structures which occur as the quadratic twist of a given torsion structure.

Let $K = \mathbb{Q}(\sqrt{-3})$ or $\mathbb{Q}(\sqrt{-1})$ and let $C_n$ denote the cyclic group of order $n$. We study how the torsion part of an elliptic curve over $K$ grows in a quadratic extension of $K$. In the case $E(K)[2] \approx C_1$ we investigate how a given torsion structure can grow in a quadratic extension and the maximum number of extensions in which it grows. We also study the torsion structures which occur as the quadratic twist of a given torsion structure. In order to achieve this we examine $N$-isogenies defined over $K$ for $N=15,20,21,24,27,30,35$.

For a fixed odd prime q we investigate the first and second order terms of the asymptotic series expansion for the number of n\le x such that q does not divide phi(n). Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the prime k-tuples conjecture and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.

We determine all Hermitian $\mathcal{O}_{\Q(\sqrt{d})}$-matrices for which every eigenvalue is in the interval [-2,2], for each d in {-2,-7,-11,-15\}. To do so, we generalise charged signed graphs to $\mathcal{L}$-graphs for appropriate finite sets $\mathcal{L}$, and classify all $\mathcal{L}$-graphs satisfying the same eigenvalue constraints. We find that, as in the integer case, any such matrix / graph is contained in a maximal example with all eigenvalues $\pm2$.

Following Hasse's example, various authors have been deriving divisibility properties of minus class numbers of cyclotomic fields by carefully examining the analytic class number formula. In this paper we will show how to generalize these results to CM-fields by using class field theory. Although we will only need some special cases, we have also decided to include a few results on Hasse's unit index for CM-fields as well, because it seems that our proofs are more direct than those given by Hasse.

Let p be an odd prime, and k_\infty the cyclotomic Z_p-extension of an abelian field k. For a finite set S of rational primes which does not include p, we will consider the maximal S-ramified abelian pro-p extension M_S(k_\infty) over k_\infty. We shall give a formula of the Z_p-rank of Gal(M_S(k_\infty)/k_\infty). In the proof of this formula, we also show that M_{q}(k_\infty)/L(k_\infty) is a finite extension for every real abelian field k and every rational prime q distinct from p, where L(k_\infty) is the maximal unramified abelian pro-p extension over k_\infty.

Let $\K$ be a galois CM extension of $\Q$ and $\K_{\infty}$ its cyclotomic $\Z_p$-extension. Let $A_n$ be the $p$-parts of the class groups in the intermediate subfields $\K_n \subset \K_{\infty}$ and $\rg{A} = \varprojlim_n A_n$. We show that the $p$-rank of $\rg{A}$ is finite, which is equivalent to the vanishing of Iwasawa's constant $\mu$ for $\rg{A}$. (Currently withdrawn)

In a previous paper, the author examined the possible torsions of an elliptic curve over the quadratic fields $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$. Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$, we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves over $\mathbb Q(i)$ and $\mathbb Q(\sqrt{-3})$.

We determine all irreducible representations of primary quasi-cyclotomic fields in this paper. The methods can be applied to determine the irreducible representations of any quasi-cyclotomic field. We also compute the Artin L-functions for a class of quasi-cyclotomic fields.

We call a (q-1)-th Kummer extension of a cyclotomic function field a quasi-cyclotomic function field if it is Galois, but non-abelian, over the rational function field with the constant field of q elements. In this paper, we determine the structure of the Galois groups of a kind of quasi-cyclotomic function fields over the base field. We also give the genus formulae of them.

We mainly study a polynomial $f_{1,n}(x)=x^{n-1} + 2x^{n-2} + 3x^{n-3} + \cdots + kx^{n-k} + \cdots + (n-1)x + n$ over $\mathbb{Z}$ and the Galois group of the minimal splitting field. First, we show that an arbitrary root $\alpha_{n}$ of $f_{1,n}(x)$ satisfies $|\alpha_{n}|\to 1$ ($n\to \infty$), and discuss the irreducibility of $f_{1,n}(x)$ over $\mathbb{Z}$ for several type $n$. After that, we show that the Galois group of $f_{1,n}(x)$ is Symmetric group $S_{n-1}$ for several type $n$. Although those roots of $f_{1,n}(x)=0$ don't draw an exact circle, it looks like a circle on complex plane. Moreover by considering that Galois groups of $f_{1,n}(x)$ are not abelian in many cases, we call such extension fields over $\mathbb{Q}$ "Non-Abelian Cycrotomic Fields" here.

Strong external difference family (SEDF) and its generalizations GSEDF, BGSEDF in a finite abelian group $G$ are combinatorial designs raised by Paterson and Stinson [7] in 2016 and have applications in communication theory to construct optimal strong algebraic manipulation detection codes. In this paper we firstly present some general constructions of these combinatorial designs by using difference sets and partial difference sets in $G$. Then, as applications of the general constructions, we construct series of SEDF, GSEDF and BGSEDF in finite fields by using cyclotomic classes.

Let K denote the quadratic field $\mathbb{Q}(\sqrt{d})$ where d=$-1$ or $-3$. Let E be an elliptic curve defined over K. In this paper, we analyze the torsion subgroups of E in the maximal elementary abelian $2$-extension of $K$.

Using an idea going back to Scholz, we construct unramified abelian extensions of cyclotomic extensions of number fields.

In this paper, we introduce a Carlitz module analogue of Mersenne primes, and prove Carlitz module analogues of several classical results concerning Mersenne primes. In contrast to the classical case, we can show that there are infinitely many composite Mersenne numbers. We also study the acquaintances of Mersenne primes including Wieferich and non-Wieferich primes in the Carlitz module context that were first introduced by Dinesh Thakur.

Discrete Fourier transforms~(DFTs) over finite fields have widespread applications in error correction coding. Hence, reducing the computational complexities of DFTs is of great significance, especially for long DFTs as increasingly longer error control codes are chosen for digital communication and storage systems. Since DFTs involve both multiplications and additions over finite fields and multiplications are much more complex than additions, recently proposed cyclotomic fast Fourier transforms (CFFTs) are promising due to their low multiplicative complexity. Unfortunately, they have very high additive complexity. Techniques such as common subexpression elimination (CSE) can be used to reduce the additive complexity of CFFTs, but their effectiveness for long DFTs is limited by their complexity. In this paper, we propose prime factor cyclotomic Fourier transforms (PFCFTs), which use CFFTs as sub-DFTs via the prime factor algorithm. When the length of DFTs is prime, our PFCFTs reduce to CFFTs. When the length has co-prime factors, since the sub-DFTs have much shorter lengths, this allows us to use CSE to significantly reduce their additive complexity. In comparison to previously proposed fast Fourier transforms, our PFCFTs achieve reduced overall complexity when the length of DFTs is at least 255, and the improvement significantly increases as the length grows. This approach also enables us to propose efficient DFTs with very long length (e.g., 4095-point), first efficient DFTs of such lengths in the literature. Finally, our PFCFTs are also advantageous for hardware implementation due to their regular structure.

I present here the proofs of results, which are obtained in my papers "On the linear forms with algebraic coefficoients of logarithms of algebraic numbers", VINITI, 1996, 1617-B96, pp. 1 - 23 (in Russian), and "On the systems of linear forms with algebraic coefficoients of logarithms of algebraic numbers", VINITI, 1996, 2663-B96, pp. 1 - 18 (in Russian).

We introduce a concept of cyclotomic association scheme C over a finite near-field. It is proved that if C is nontrivial, then Aut(C)

For a real abelian field and for an odd prime p splitting in the field, we study a map between the p-parts of the class group and of the quotient of units modulo Cyclotomic Units, respectively, along the cyclotomic Z_p-extension of the field. We determine the kernel and the cokernel of this map assuming Greenberg's conjecture on the vanishing of the lambda-invariant of the extension.

Let $\K$ be a finite extension of a characteristic zero field $\F$. We say that the pair of $n\times n$ matrices $(A,B)$ over $\F$ represents $\K$ if $\K \cong \F[A]/ < B >$ where $\F[A]$ denotes the smallest subalgebra of $M_n(\F)$ containing $A$ and $ < B >$ is an ideal in $\F[A]$ generated by $B$. In particular, $A$ is said to represent the field $\K$ if there exists an irreducible polynomial $q(x)\in \F[x]$ which divides the minimal polynomial of $A$ and $\K \cong \F[A]/ < q(A) >$. In this paper, we identify the smallest circulant-matrix representation for any subfield of a cyclotomic field. Furthermore, if $p$ is any prime and $\K$ is a subfield of the $p$-th cyclotomic field, then we obtain a zero-one circulant matrix $A$ of size $p\times p$ such that $(A,\J)$ represents $\K$, where $\J$ is the matrix with all entries 1. In case, the integer $n$ has at most two distinct prime factors, we find the smallest 0-1 companion-matrix that represents the $n$-th cyclotomic field. We also find bounds on the size of such companion matrices when $n$ has more than two prime factors.

Let $\K$ be a finite extension of a characteristic zero field $\F$. We say that the pair of $n\times n$ matrices $(A,B)$ over $\F$ represents $\K$ if $\K \cong \F[A]/ < B >$ where $\F[A]$ denotes the smallest subalgebra of $M_n(\F)$ containing $A$ and $ < B >$ is an ideal in $\F[A]$ generated by $B$. In particular, $A$ is said to represent the field $\K$ if there exists an irreducible polynomial $q(x)\in \F[x]$ which divides the minimal polynomial of $A$ and $\K \cong \F[A]/ < q(A) >$. In this paper, we identify the smallest circulant-matrix representation for any subfield of a cyclotomic field. Furthermore, if $p$ is any prime and $\K$ is a subfield of the $p$-th cyclotomic field, then we obtain a zero-one circulant matrix $A$ of size $p\times p$ such that $(A,\J)$ represents $\K$, where $\J$ is the matrix with all entries 1. In case, the integer $n$ has at most two distinct prime factors, we find the smallest 0-1 companion-matrix that represents the $n$-th cyclotomic field. We also find bounds on the size of such companion matrices when $n$ has more than two prime factors.

We say a tame Galois field extension $L/K$ with Galois group $G$ has trivial Galois module structure if the rings of integers have the property that $\Cal{O}_{L}$ is a free $\Cal{O}_{K}[G]$-module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes $l$ so that for each there is a tame Galois field extension of degree $l$ so that $L/K$ has nontrivial Galois module structure. However, the proof does not directly yield specific primes $l$ for a given algebraic number field $K.$ For $K$ any cyclotomic field we find an explicit $l$ so that there is a tame degree $l$ extension $L/K$ with nontrivial Galois module structure.

Cyclotomic fast Fourier transforms (CFFTs) are efficient implementations of discrete Fourier transforms over finite fields, which have widespread applications in cryptography and error control codes. They are of great interest because of their low multiplicative and overall complexities. However, their advantages are shown by inspection in the literature, and there is no asymptotic computational complexity analysis for CFFTs. Their high additive complexity also incurs difficulties in hardware implementations. In this paper, we derive the bounds for the multiplicative and additive complexities of CFFTs, respectively. Our results confirm that CFFTs have the smallest multiplicative complexities among all known algorithms while their additive complexities render them asymptotically suboptimal. However, CFFTs remain valuable as they have the smallest overall complexities for most practical lengths. Our additive complexity analysis also leads to a structured addition network, which not only has low complexity but also is suitable for hardware implementations.

Rosen M. gave a determinant formula for relative class numbers for the P-th cyclotomic function fields in the case of the monic irreducible polynomial P, which is regarded as an analogue of the classical Maillet determinant. In this paper, we will give a determinant formula for the relative congruence zeta functions for cyclotomic function fields. Our formula is regarded as a generalization of the determinant formula for the relative class number.

1Lilac Fairy Book

By Andrew Lang

Andrew Lang's Fairy Books or Andrew Lang's "Coloured" Fairy Books are a twelve-book series of fairy tale collections. Although Andrew Lang did not collect the stories himself from the oral tradition, the extent of his sources (who had collected them originally), made them an immensely influential collection, especially as he used foreign-language sources, giving many of these tales their first appearance in English. As acknowledged in the prefaces, although Lang himself made most of the selections, his wife and other translators did a large portion of the translating and telling of the actual stories. Many of them were illustrated by Henry J. Ford. Lancelot Speed also did some illustrations. (Summary from Wikipedia)

2Blue Fairy Book

By Andrew Lang

Andrew Lang's Blue Fairy Book (1889) was a beautifully produced and illustrated edition of fairy tales that has become a classic. This was followed by many other collections of fairy tales, collectively known as Andrew Lang's Fairy Books. (Wikipedia)

3Yellow Fairy Book

By Andrew Lang

"Andrew Lang's Fairy Books or Andrew Lang's "Coloured" Fairy Books are a twelve-book series of fairy tale collections. Although Andrew Lang did not collect the stories himself from the oral tradition, the extent of his sources (who had collected them originally), made them an immensely influential collection, especially as he used foreign-language sources, giving many of these tales their first appearance in English. As acknowledged in the prefaces, although Lang himself made most of the selections, his wife and other translators did a large portion of the translating and telling of the actual stories." (summary from Wikipedia)

4Red Fairy Book

By Andrew Lang

<i>The Red Fairy Book</i> is the second in a series of twelve books known as <i>Andrew Lang's Fairy Books</i> or <i>Andrew Lang's "Coloured" Books</i>. The series was immensely popular and proved of great influence in children's literature, increasing the popularity of fairy tales over tales of real life. (Summary adapted from Wikipedia)

5Green Fairy Book

By Andrew Lang

Andrew Lang's Fairy Books or Andrew Lang's "Coloured" Fairy Books constitute a twelve-book series of fairy tale collections. Although Andrew Lang did not collect the stories himself from the oral tradition, the extent of his sources, who had collected them originally (with the notable exception of Madame d'Aulnoy), made them an immensely influential collection, especially as he used foreign-language sources, giving many of these tales their first appearance in English. As acknowledged in the prefaces, although Lang himself made most of the selections, his wife and other translators did a large portion of the translating and telling of the actual stories.<br><br>"The irony of Lang's life and work is that although he wrote for a profession—literary criticism; fiction; poems; books and articles on anthropology, mythology, history, and travel ... he is best recognized for the works he did not write."<br /><br /> Many of the books were illustrated by Henry J. Ford, Lancelot Speed, and G. P. Jacomb-Hood also contributed some illustrations.

6Crimson Fairy Book

By Andrew Lang

The Crimson Fairy Book contains thirty-six stories collected from around the world and edited by Andrew Lang. Many tales in this book are translated, or adapted, from those told by mothers and nurses in Hungary; others are familiar to Russian nurseries; the Servians are responsible for some; a rather peculiarly fanciful set of stories are adapted from the Roumanians; others are from the Baltic shores; others from sunny Sicily; a few are from Finland, and Iceland, and Japan, and Tunis, and Portugal. No doubt many children will like to look out these places on the map, and study their mountains, rivers, soil, products, and fiscal policies, in the geography books. The peoples who tell the stories differ in colour; language, religion, and almost everything else; but they all love a nursery tale. The stories have mainly been adapted or translated by Mrs. Lang, a few by Miss Lang and Miss Blackley. (Summary from the preface)

7Short History of Scotland

By Andrew Lang

A Short History of Scotland is a concise introduction to the history of Scotland from Roman times to the last Jacobite rebellion, written by the author of a much longer Scottish history. (Summary by Sibella Denton.)

8Arabian Nights Entertainments

By Andrew Lang

The Arabian Nights is a collection of Perso-Arabic folk tales and other stories. The collection, or at least certain stories drawn from it (or purporting to be drawn from it), became widely known in the West from the 18th century, after it was translated from the Arabic — first into French and then into English and other European languages. The first English language edition, based on Galland's French rather than the original Arabic, rendered the title as The Arabian Nights' Entertainment - and this, or simply The Arabian Nights, has been the title by which it has been best known to English-speaking people ever since. (Summary from Wikipedia)

9Antiquary

By Andrew Lang and Sir Walter Scott

Illegitimacy, false identity, and bankruptcy are the major elements of Sir Walter Scott's 1816 novel, The Antiquary. Set in the period of the French Revolution, the novel's hero, Lovel, struggles to gain repute and the hand of his beloved despite his uncertain parentage. During these pursuits, he befriends the title's antiquary, Johnathan Oldbuck, who finds Lovel a captive audience to his scholarly studies and a tragic likeness to his own disappointments in love. Readers will discover whether Lovel's acts of bravery and courage ultimately earn him the birth and fortunes of a nobleman. (Summary by S. Kovalchik)

10Orange Fairy Book

By Andrew Lang

Andrew Lang's Fairy Books or Andrew Lang's "Coloured" Fairy Books constitute a twelve-book series of fairy tale collections. Although Andrew Lang did not collect the stories himself from the oral tradition, the extent of his sources, who had collected them originally (with the notable exception of Madame d'Aulnoy), made them an immensely influential collection, especially as he used foreign-language sources, giving many of these tales their first appearance in English. As acknowledged in the prefaces, although Lang himself made most of the selections, his wife and other translators did a large portion of the translating and telling of the actual stories.<br><br>Many of the books were illustrated by Henry J. Ford. Lancelot Speed, and G. P. Jacomb Hood also contributed some illustrations. (Summary by Wikipedia)

11Pink Fairy Book

By Andrew Lang

All people in the world tell nursery tales to their children, and the stories are apt to be like each other everywhere. A child who has read the Blue and Red and Yellow Fairy Books will find some old friends with new faces in the Pink Fairy Book. Courage, youth, beauty, kindness, have many trials, but they always win the battle; while witches, giants, unfriendly cruel people, are on the losing hand. So it ought to be, and so, on the whole, it is and will be; and that is all the moral of fairy tales. We cannot all be young, alas! and pretty, and strong; but nothing prevents us from being kind, and no kind man, woman, or beast or bird, ever comes to anything but good in these oldest fables of the world. (Summary of the Author's Preface by Elliott Miller)

12He

By Andrew Lang and Walter Herries Pollock

This book is a parody of the famous swashbuckling novel, She, by H. Rider Haggard. (summary by Neeru Iyer)<br><br>You will not think, therefore, that the respectful Liberty we have taken with your Wondrous Tale... indicates any lack of Loyalty to our Lady Ayesha.<br><br>Her beauties are beyond the reach of danger from Burlesque, nor does her form flit across our humble pages. (taken from the Dedication)<br /><br /> The characters of Editor and Publisher, who make comments throughout the story were read by: Editor – <a href="http://librivox.org/reader/3588">Neeru Iyer</a> and Publisher – <a href="http://librivox.org/reader/2506">Julie VW</a>

13Grey Fairy Book

By Andrew Lang

The tales in the Grey Fairy Book are derived from many countries — Lithuania, various parts of Africa, Germany, France, Greece, and other regions of the world. They have been translated and adapted by Mrs. Dent, Mrs. Lang, Miss Eleanor Sellar, Miss Blackley, and Miss Hang. 'The Three Sons of Hali' is from the last century 'Cabinet des Fees,' a very large collection. The French author may have had some Oriental original before him in parts; at all events he copied the Eastern method of putting tale within tale, like the Eastern balls of carved ivory. The stories, as usual, illustrate the method of popular fiction. A certain number of incidents are shaken into many varying combinations, like the fragments of coloured glass in the kaleidoscope. Probably the possible combinations, like possible musical combinations, are not unlimited in number, but children may be less sensitive in the matter of fairies than Mr. John Stuart Mill was as regards music. (Summary from the preface)

This is a children's version of Jonathan Swift's novel <i>Gulliver's Travels</i>, from the <i>Told to the Children Series</i> (published in 1910). The children's adventure story covers Gulliver's visits to the lands of Lilliput and Brobdingnag. (Summary by Laurie Anne Walden)

15Olive Fairy Book

By Andrew Lang

Andrew Lang’s Olive Fairy Book (1907) was a beautifully produced and illustrated edition of fairy tales that has become a classic. This was followed by many other collections of fairy tales, collectively known as Andrew Lang’s Fairy Books. (Summary by Wikipedia)

16Brown Fairy Book

By Andrew Lang

Andrew Lang’s Brown Fairy Book (1904) was a beautifully produced and illustrated edition of fairy tales that has become a classic. This was one of many other collections of fairy tales, collectively known as Andrew Lang’s Fairy Books. (Summary by Wikipedia)

17Violet Fairy Book

By Andrew Lang

Andrew Lang’s Violet Fairy Book (1901) was a beautifully produced and illustrated edition of fairy tales that has become a classic. This was one of many other collections of fairy tales, collectively known as Andrew Lang’s Fairy Books. (Summary by Wikipedia)

Joan of Arc is a folk heroine of France and a Roman Catholic saint. Claiming divine guidance, she led the French army to several important victories during the Hundred Years' War, which paved the way for the coronation of Charles VII of France. Captured by her enemies, she was sold to the English and put on trial for charges of "insubordination and heterodoxy". She was burned at the stake for heresy when she was 19 years old. Twenty-five years after her execution, an inquisitorial court examined the trial, pronounced her innocent, and declared her a martyr. Joan of Arc was beatified in 1909 and canonized in 1920. She is one of the patron saints of France. Most of what is known today of her story comes from the detailed court records of her trials. (Summary by timothyFR)

19Letters to Dead Authors

By Andrew Lang

With twenty two letters, addressed to various already deceased authors, Andrew Lang discusses literary subjects with his usual humour and acidity. The impulse for the writing of the letters came, almost as a joke, from the editor of the 'St. James's Gazette,' and sixteen of the letters collected in the volume appeared first in that journal. According to the author, "some of the Letters are written rather to suit the Correspondent than to express the writer's own taste or opinions"; in all of them, though, the reader will find the charming wit of their author. (Summary by Leni)

LibriVox volunteers bring you 16 readings of <em>Romance</em> by Andrew Lang, probably best known as Edward Elgar's song <em>My Love Dwelt in a Northern Land</em>. Interestingly, Lang initially refused permission for his words to be used as lyrics, and Elgar's wife Alice wrote alternative words <em>Afar, amidst the Sunny Isles</em> for the song. However, Lang later relented and gave permission for his poem to be used. <br/><br />The poem was initially published in The Century Magazine, May 1882, and this is the version recorded here. Later collections of Lang's poetry omit the third verse. (Summary by Ruth Golding)

21Stories from the Faerie Queen

By Jean Lang and Edmund Spenser

A major work by Spenser, The Faerie Queen, was published between 1590 and 1596. As an allegorical work, it can be read on many levels. According to Jeanie Lang, Spenser always looked for the beautiful and the good when he wrote. Lang said, "There are many stories in The Faerie Queen, and out of these all I have told you only eight." The eight are "Una and the Lion," "St. Gergoe and the Dragon," "Britomart and the Magic Mirror," "The Quest of Sir Gregory," "Pastorella," "Cambell and Triamond," "Marinell the Sea-Nymph's Son," and "Flormell and the Witch." - Summary by Bill Boerst

22Old Friends, Essays in Epistolary Parody

By Andrew Lang

"Every fancy which dwells much with the unborn and immortal characters of Fiction must ask itself, Did the persons in contemporary novels never meet? In so little a world their paths must often have crossed, their orbits must have intersected, though we hear nothing about the adventure from the accredited narrators. In historical fiction authors make their people meet real men and women of history—Louis XI., Lazarus, Mary Queen of Scots, General Webbe, Moses, the Man in the Iron Mask, Marie Antoinette; the list is endless. But novelists, in spite of Mr. Thackeray’s advice to Alexandre Dumas, and of his own example in “Rebecca and Rowena,” have not introduced each other’s characters" (from the Introduction). In this volume, Andrew Lang shows, what a letter from one fictional character to another might look like.

23Tales of King Arthur and the Round Table (version 2)

By Andrew Lang

The tales of King Arthur and his Knights are of Celtic origin. The Celts were the people who occupied Britain at the time when the history of the country opens, and a few words are necessary to explain why the characters in the stories act and speak as though they belonged to a later age. These stories are adapted from the Book of Romance by ANDREW LANG.<br /><br /> It is believed that King Arthur lived in the sixth century, just after the Romans withdrew from Britain, and when the Britons, left to defend themselves against the attacks of the marauding Saxons, rose and defeated them at Mount Badon, securing to themselves peace for many years. It was probably about this time that King Arthur and his company of Knights performed the deeds which were to become the themes of stories and lays for generations afterwards.<br /><br /> In olden times, it was the custom of minstrels and story-tellers to travel through the land from court to court, telling of tales of chivalry and heroism, and for many centuries the tales of King Arthur formed the stock from which the story-teller drew.<br /><br /> In this way the stories came to be handed down from father to son, in Brittany (whose people are of the same family as the Welsh) as well as in Wales and England, and by this means alone were they prevented from being lost. But in the reigns of Henry II. and Richard I., they were set down on paper, and so became literature. Before this, however, a British writer had written out some of the tales, and from him as well as from the lips of the bards and story-tellers of their own generation, the writers in the time of Henry II. were able to collect their information. One of the most famous of the books compiled by old English writers was the “Historia Britonum,” which was written (in Latin) by Geoffrey, Bishop of Asaph. It contained an account of a war which King Arthur waged in Western Europe, but made no mention of the Holy Grail.<br /><br /> From this and other books of romances compiled in England, and very largely, too, from books of French romances, Sir Thomas Malory obtained the material for his “Morte d’Arthur,” which was written in 1470. This is the most famous of the early books of Arthurian legend, and it is from the “Morte d’Arthur” that most of the stories in this book are taken. Some, however, are taken from the “High History of the Holy Graal,” translated from the French by Dr. Sebastian Evans. The language throughout has been modified with a view to making the legends more easy of study. - Summary by The introduction and BC

24Tales Of King Arthur And The Round Table

By Andrew Lang

The tales of King Arthur and his Knights are of Celtic origin. The Celts were the people who occupied Britain at the time when the history of the country opens… It is believed that King Arthur lived in the sixth century, just after the Romans withdrew from Britain… the stories came to be handed down from father to son, in Brittany (whose people are of the same family as the Welsh) as well as in Wales and England… [story-tellers altered the stories to suit their times down through the centuries] …and so in their altered and historically inaccurate form they have reached us at the present day. …Sir Thomas Malory obtained the material for his “Morte d’Arthur,” which was written in 1470. This is the most famous of the early books of Arthurian legend, and it is from the “Morte d’Arthur” that most of the stories in this book are taken…. The language throughout has been modified with a view to making the legends more easy of study. - Summary by Book Introduction and david wales

25Tales of Troy: Ulysses the Sacker of Cities

By Andrew Lang

These are short stories about the life of Ulysses, the stealing of Helen, Paris, battles, Trojan horses, and more! - Summary by Maggie Travers

26Strange Story Book

By Leonora Blanche Lang

A collection of stories compiled by the widow of the renowned children's short-story collector, Andrew Lang. She describes this as the final installment of his series of fairy stories and includes some of those stories he had not included in other collections. In the preface, she pays tribute to her husband, saying he liked old things more than new, he loved all beasts' because they were beasts', fairies, cricket and dancing. He longed to visit far away places, but hated to travel. But most of all he befriended children wherever he went and they shared stories with him. Only his own children mocked him. - Summary by Lynne Thompson

27Strange Story Book (version 2)

By Leonora Blanche Lang

This is the last book in the The Andrew Lang "Color" Fairy Books series published after his death by his wife. She wrote the stories and he had edited them before his demise. It is a collection of stories that are a bit different than the Fairy Books: still colorful and well written but telling tales about decidedly odd topics. A word of <font color="red">Caution: </font> please read the stories before your children do, some of them contain some violent imagery. - Summary by Linda Andrus

28Summer's Ending

By Andrew Lang

Andrew Lang was a Scottish poet, novelist, literary critic, and contributor to the field of anthropology. He is best known as a collector of folk and fairy tales. The Andrew Lang lectures at the University of St Andrews are named after him. - Summary by Wikipedia

29World's Desire

By H. Rider Haggard and Andrew Lang

H. Rider Haggard (author of King Solomon's Mines the Allan Quatermain Series, and many more) and Andrew Lang (author of, among others, the rainbow coloured fairytale books) collaborate to lend their talent to one of the most fascinating and well known stories of all times. Odysseus returns home from the war, but does not find the peace and quiet which he craves. His home is ravaged, and his wife Penelope is dead. He comunicates with an old flame, the beautiful Helen of Troy, who sends him to his ultimate and defining last journey. Read about his adventures, and what might have been, in this beautiful novel by two of Britian's best Vctorian novelists. - Summary by Stav Nisser and Wikipedia.

30Book of Myths

By Jean Lang

This is a collection of myths--mostly Greek with a smattering of others from the east--written in a clear and easy-to-read style. Lang complemented each myth with poetry by other authors who, like her, were inspired by these ancient stories of the gods. Lang chose these stories because they portrayed heroic gods, faithfully and blindly worshipped by man. Ultimately, however, these gods demonstrated the same frailties as humans, and were found to be just as corrupt. Still, as Lang said, in spite of this these myths portrayed "a wonderful humanity that strikes a vibrant cord . . . ." This is significant to a deeper understanding of the collection as it was published in 1914 against the backdrop of the first world war, the war to end all wars--a war that doomed millions of common men to suffer "Promethean agonies," and die on battlefields in a most un-heroic way. As you listen to the narration, compare the gods of myth--with all their human frailties--to the 20th century, god-like European leaders who traded the wonderful innocence of humanity for the notion of "a noble cause." - Summary by James K. White