Primitive Pth Root Research Articles

Frattini-resistant direct products of pro-p groups

A pro-p group G is called strongly Frattini-resistant if the function H↦Φ(H), from the poset of all closed subgroups of G into itself, is a poset embedding. Frattini-resistant pro-p groups appear naturally in Galois theory. Indeed, every maximal pro-p Galois group over a field that contains a primitive pth root of unity (and also contains −1 if p=2) is strongly Frattini-resistant. Let G1 and G2 be non-trivial pro-p groups. We prove that G1×G2 is strongly Frattini-resistant if and only if one of the direct factors G1 or G2 is torsion-free abelian and the other one has the property that all of its closed subgroups have torsion-free abelianization. As a corollary we obtain a group theoretic proof of a result of Koenigsmann on maximal pro-p Galois groups that admit a non-trivial decomposition as a direct product. In addition, we give an example of a group that is not strongly Frattini-resistant, but has the property that its Frattini-function defines an order self-embedding of the poset of all topologically finitely generated subgroups.

The Cassels heights of cyclotomic integers

We study the set {mathscr {C}} of mean square values of the moduli of the conjugates of all nonzero cyclotomic integers beta . For its kth derived set {mathscr {C}}^{(k)}, we show that {mathscr {C}}^{(k)}=(k+1){mathscr {C}},, (kge 0), so that also {mathscr {C}}^{(k)}+{mathscr {C}}^{(ell )}={mathscr {C}}^{(k+ell +1)},,(k,ell ge 0). Furthermore, we describe precisely the restricted set {mathscr {C}}_p where the beta are confined to the ring {mathbb {Z}}[omega _p], where p is an odd prime and omega _p is a primitive pth root of unity. In order to do this, we prove that both of the quadratic polynomials a^2+ab+b^2+c^2+a+b+c and a^2+b^2+c^2+ab+bc+ca+a+b+c are universal.

The Yagita Invariant of Symplectic Groups of Large Rank

Fix a prime p, and let {mathcal {O}} denote a subring of {mathbb {C}} that is either integrally closed or contains a primitive pth root of 1. We determine the Yagita invariant at the prime p for the symplectic group {mathrm{Sp}}(2n,{mathcal {O}}) for all nge p-1.

Hopf-Galois module structure of tamely ramified radical extensions of prime degree

Let K be a number field and let L/K be a tamely ramified radical extension of prime degree p. If K contains a primitive pth root of unity then L/K is a cyclic Kummer extension; in this case the group algebra K[G] (with G=Gal(L/K)) gives the unique Hopf-Galois structure on L/K, the ring of algebraic integers OL is locally free over OK[G] by Noether's theorem, and Gómez Ayala has determined a criterion for OL to be a free OK[G]-module. If K does not contain a primitive pth root of unity then L/K is a separable, but non-normal, extension, which again admits a unique Hopf-Galois structure. Under the assumption that p is unramified in K, we show that OL is locally free over its associated order in this Hopf-Galois structure and determine a criterion for it to be free. We find that the conditions that appear in this criterion are identical to those appearing in Gómez Ayala's criterion for the normal case.

On the Chebotarëv theorem over finite fields

Let p be a prime number and let Vp be the Vandermonde matrix with (i,j)-entry equal to ωij for 0≤i,j≤p−1, where ω is a primitive pth root of unity in the complex field. The classical Chebotarëv theorem says that all square submatrices of Vp have nonzero determinant. In this paper, we establish the Chebotarëv theorem over finite fields by imposing certain conditions.

Galois groups of some iterated polynomials over cyclotomic extensions

Let \(\varphi _p(z)=(z-1)^p+2-\zeta _p\), where \(\zeta _p\in \bar{\mathbb {Q}}\) is a primitive pth root of unity. Building on previous work, we show that the nth iterate \(\varphi _p^n(z)\) has Galois group \([C_p]^n\), an iterated wreath product of cyclic groups, whenever p is not a Wieferich prime.

On units in loop algebra $$F[M(Dih(C_p^2),2)]$$ F [ M ( D i h ( C p 2 ) , 2 )

Let $$L=M(G,2)$$ be a Moufang loop, F[L] be its loop algebra over a field F and J(F[L]) be the Jacobson radical of F[L]. In this article, we obtain the unit loop of F[L] / J(F[L]) and the structure of $$1+J(F[L])$$ , where $$L=M(G,2)$$ is obtained from the generalized dihedral group $$G=Dih(C_p^2)$$ , p an odd prime and F is a finite field of characteristic 2 containing a primitive pth root of unity.

Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups

Let F be a finite field of characteristic p and K a field which contains a primitive pth root of unity and char K ≠ p. Suppose that a classical group G acts on the F-vector space V. Then it can induce the actions on the vector space V V and on the group algebra K[V V], respectively. In this paper we determine the structure of G-invariant ideals of the group algebra K[V V], and establish the relationship between the invariant ideals of K[V] and the vector invariant ideals of K[V V], if G is a unitary group or orthogonal group. Combining the results obtained by Nan and Zeng (2013), we solve the problem of vector invariant ideals for all classical groups over finite fields.

Ramification of the Kummer extension generated from torsion points of elliptic curves

For a prime p, let ζp denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field k with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over k(ζp). In this paper, for p = 5 and 7, we study the ramification of such Kummer extensions using explicit Kummer generators directly computed by Verdure in 2006.

Relative Brauer group and pro-p Galois group of pre-p-henselian fields

Let p be a prime number and (F, v) a valued field. In this paper, we find a presentation for the p-torsion part of the Brauer group Br (F), by means of the valuation v. We only assume that F has a primitive pth root of the unity and the residue class field has characteristic not equal to p. This result naturally leads to consider valued fields that we call pre-p-henselian fields. It concerns valuations compatible with Rp, the p-radical of the field. To be precise, Rp is the radical of the skew-symmetric pairing which associates to a pair (a, b) the class of the symbol algebra (F; a, b) in Br F. In our main result, we state that pre-p-henselian fields are precisely the fields for which the Galois group of the maximal Galois p-extension admits a particular decomposition as a free pro-p product.

Indices of inseparability and refined ramification breaks

Let K be a finite extension of Qp which contains a primitive pth root of unity ζp. Let L/K be a totally ramified (Z/pZ)2-extension which has a single ramification break b. In [2] Byott and Elder defined a “refined ramification break” b⁎ for L/K. In this paper we prove that if p>2 and the index of inseparability i1 of L/K is not equal to p2b−pb then b⁎=i1−p2b+pb+b.

Faddeev invariants for central simple algebras over rational function fields

Let p be a prime, k a field, containing a primitive pth root of unity, char k ≠ p. We give an upper bound for the Faddeev index of a central simple algebra of exponent p over the rational function field k(t) in the case where the ramification set of the algebra consists of rational points. This bound depends only on the number of ramification points and in certain cases turns out to be strict. In the case where p = 2 and the ramification set in \({\mathbb{A}_k^1}\) consists of three rational points we compute the Faddeev index, using the language of quadratic forms. Let X be a smooth geometrically irreducible complete curve over k. We show that there exist algebras of exponent p over k(X) with the prescribed Faddeev index, provided there are algebras of exponent p and arbitrarily large index over k. In the last section of the paper we consider another invariant of a central simple algebra of prime exponent p over k(t), the so called Faddeev cyclic length. In certain cases we compute this invariant, using triviality of the divided power operations on central simple cyclic algebras of exponent p.

The Quadratic Gauss Sum Redux

Let p be an odd prime and ζ be a primitive pth-root of unity. For any integer a prime to p, let denote the Legendre symbol, which is 1 if a is a square mod p, and is -1 otherwise. Using Euler's Criterion that mod p, it follows that the Legendre symbol gives a homomorphism from the multiplicative group of nonzero elements Fp* of Fp Z/pZ to {±1}. Gauss's law of quadratic reciprocity states that for any other odd prime q,

KUMMER GENERATORS AND TORSION POINTS OF ELLIPTIC CURVES WITH BAD REDUCTION AT SOME PRIMES

For a prime p, let ζp denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field K with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over K(ζp), and in this paper, we study the ramification of such Kummer extensions using the Kummer generators directly computed by Verdure in 2006. For quadratic fields K, we also give unramified Kummer extensions over K(ζp) generated from elliptic curves over K having a p-torsion point with bad reduction at certain primes. Many of these unramified Kummer extensions have not appeared in the previous work using fundamental units of quadratic fields.

On The Relative Class Number of Special Cyclotomic Fields

Let p be an odd prime, p be a primitive pth root of unity and h p be the relative class number of the pth cyclotomic eld Q( p) over the rationals Q dened by p. The main purpose of this paper is to discuss arithmetic properties of factors of hp for an odd prime p of the form p = 4q + 1 with q a prime.

Some properties of the supersoluble formation and the supersoluble residual of a group

Purpose: In this paper, We determine the finite group G = HK such that K is a supersoluble subgroup of G ,a ndH is not a supersoluble subgroup of G. Methods: Let p, q, r be primes such that p < q < r ,a ndp, q are not a divisor of r − 1, and p is not a divisor of q − 1. Let X be a group of order p, and let F = GF(q) and L = GF(r) such that the filed F contains a primitive pth root of

Trace Representation and Linear Complexity of Binary $e$th Power Residue Sequences of Period $p$

Let p = ef + 1 be an odd prime for some e and e and let f, be the finite field with Fp elements. In this paper, we explicitly describe the trace representations of the binary characteristic sequences (of period p) of all the cyclic difference sets D which are some union of cosets of eth powers He in Fp* (=Δ Fp\{0}) for e ≤ 12. For this, we define eth power residue sequences of period p, which include all the binary characteristic sequences mentioned above as special cases, and reduce the problem of determining their trace representations to that of determining the values of the generating polynomials of cosets of He in Fρ* at some primitive pth root of unity, and some properties of these values are investigated. Based on these properties, the trace representation and linear complexity not only of the characteristic sequences of all the known eth residue difference sets, but of all the sixth power residue sequences are determined. Furthermore, we have determined the linear complexity of a nonconstant eth power residue sequence for any e to be either p - 1 or p whenever (e, (p-1)/n) = 1, where n is the order of 2 mod p.

Galois module structure of Galois cohomology for embeddable cyclic extensions of degree p n

Let p>2 be prime, and let n,m be positive integers. For cyclic field extensions E/F of degree p^n that contain a primitive pth root of unity, we show that the associated F_p[Gal(E/F)]-modules H^m(G_E,mu_p) have a sparse decomposition. When E/F is additionally a subextension of a cyclic, degree p^{n+1} extension E'/F, we give a more refined F_p[Gal(E/F)]-decomposition of H^m(G_E,mu_p).

On the Weight Distributions of Two Classes of Cyclic Codes

Let q=pm where p is an odd prime, mges2, and 1lesklesm-1. Let Tr be the trace mapping from Fq to Fp and zetap=e2pii/p be a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: SigmaxisinF qchi(alphaxp k +1+betax2) (alpha, betaisinFq) where chi(x)=zetap Tr(x) is the canonical additive character of Fq. As applications, we have the following. 1) We determine the weight distribution of the cyclic codes C1 and C2 over Fpt with parity-check polynomial h2(x)h3(x) and h1(x)h2(x)h3(x), respectively, where t is a divisor of d=gcd(m, k), and h1(x), h2(x) , and h3(x) are the minimal polynomials of pi-1, pi-2, and pi-(p k +1) over Fpt, respectively, for a primitive element pi of Fq. 2) We determine the correlation distribution between two m-sequences of period q-1. Moreover, we find a new class of p-ary bent functions. This paper extends the results in Feng and Luo (2008).

Endomorphisms of superelliptic jacobians

Let K be a field of characteristic zero, n ≥ 5 an integer, f(x) an irreducible polynomial over K of degree n, whose Galois group contains a doubly transitive simple non-abelian group. Let p be an odd prime, \({\mathbb{Z}[\zeta_p]}\) the ring of integers in the pth cyclotomic field, Cf, p : yp = f(x) the corresponding superelliptic curve and J(Cf, p) its jacobian. Assuming that either n = p + 1 or p does not divide n(n − 1), we prove that the ring of all endomorphisms of J(Cf, p) coincides with \({\mathbb{Z}[\zeta_p]}\) . The same is true if n = 4, the Galois group of f(x) is the full symmetric group S4 and K contains a primitive pth root of unity.