共查询到20条相似文献,搜索用时 31 毫秒
1.
Circulant matrices are used to construct polynomials, associated with Chebyshev polynomials of the first kind, whose roots are real and made explicit. Then the Galois groups of the polynomials are computed, giving rise to new examples of polynomials with cyclic Galois groups and Galois groups of order p(p−1) that are generated by a cycle of length p and a cycle of length p−1. 相似文献
2.
Let E/F be a Galois extension of number fields with Galois group G=Gal(E/F), and let p be a prime not dividing #G. In this paper, using character theory of finite groups, we obtain the upper bound of #K2OE if the group K2OE is cyclic, and prove some results on the divisibility of the p-rank of the tame kernel K2OE, where E/F is not necessarily abelian. In particular, in the case of G=Cn, Dn, A4, we easily get some results on the divisibility of the p-rank of the tame kernel K2OE by the character table. Let E/Q be a normal extension with Galois group Dl, where l is an odd prime, and F/Q a non-normal subextension with degree l. As an application, we show that f|p-rank K2OF, where f is the smallest positive integer such that pf≡±1(mod l). 相似文献
3.
A useful criterion characterizing a monic irreducible polynomial over with Galois group Dp (the dihedral group of order 2p, p: prime) is given by making use of the geometry of Dp, i.e., Dp is the symmetry group of the regular p-gon. We derive explicit numerical examples of polynomials with dihedral Galois groups D5 and D7. 相似文献
4.
J. E. Humphreys 《代数通讯》2013,41(6):475-490
With a Weyl group W and a positive integer p are associated p-linkage classes of weights [4,13]. Small deformations of such classes by elements of W are introduced here. These lead in turn to certain polynomials in p with highest term pm, m = number of positive roots (one polynomial for each conjugacy class of W), which are written down explicitly for types A1, A2, B2. These polynomials give (for each prime p) the degrees of the various large series of irreducible characters of the corresponding Chevalley group over the field of p elements. Indeed, the formal behavior of weights appears to reflect the actual behavior of the characters under reduction modulo p. 相似文献
5.
David E. Rohrlich 《Journal of Number Theory》2004,107(1):8-24
For a prime p?7 the pth roots of certain modular units are shown to generate the second layer of the extension of function fields cut out by the universal Galois deformation of the representation on p-division points of a universal elliptic curve. It follows that certain Galois representations obtained by specializing the modular invariant to a rational number have large image. 相似文献
6.
《Journal of Number Theory》1986,24(3):360-372
Let K be a real algebraic number field. Suppose that G occurs as a Galois group of a normal real extension field of K. Using elementary methods, we show that certain types of split extensions of an elementary abelian 2-group by G also occur as Galois groups of normal real extensions of K. Among other examples, we show that Sylow 2-subgroups of the symmetric and alternating groups of degree 2n, as well as the Weyl groups of type Bn and Dn, occur as Galois groups of real extensions of the rationals. 相似文献
7.
G. Kuba 《Archiv der Mathematik》2005,85(1):70-78
Let A
p
⊂ C denote the set of all algebraic numbers such that α ∈ A
p
if and only if α is a zero of a (not necessarily irreducible) polynomial with positive rational coefficients. We give several results concerning the numbers in A
p
. In particular, the intersection of A
p
and the unit circle |z| = 1 is investigated in detail. So we determine all numbers of degree less than 6 on the unit circle which lie in the set A
p
. Further we show that when α is a root of an irreducible rational polynomial p(X) of degree ≠ 4 whose Galois group contains the full alternating group, α lies in A
p
if and only if no real root of p(X) is positive.Received: 19 November 2004; revised: 9 February 2005 相似文献
8.
We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ
p
[X] is the value at p of a rational function φ
n
(X)∈ℚ(X). All rational functions involved are effectively computable.
Received: 15 September 1998 / Revised version: 21 October 1999 相似文献
9.
For an odd prime p we construct an infinite class of non-isomorphic Hopf algebras of dimension p
4 over an infinite field containing primitive p-th roots of unity, answering in the negative a long standing conjecture of Kaplansky.
Oblatum 6-XI-1997 / Published online: 12 November 1998 相似文献
10.
James P. Cossey 《Algebras and Representation Theory》2011,14(2):391-402
The Alperin weight conjecture states that if G is a finite group and p is a prime, then the number of irreducible Brauer characters of a group G should be equal to the number of conjugacy classes of p-weights of G. This conjecture is known to be true for the symmetric group S
n
, however there is no explicit bijection given between the two sets. In this paper we develop an explicit bijection between
the p-weights of S
n
and a certain set of partitions that is known to have the same cardinality as the irreducible Brauer characters of S
n
. We also develop some properties of this bijection, especially in relation to a certain class of partitions whose corresponding
Specht modules over fields of characteristic p are known to be irreducible. 相似文献
11.
Jürgen G. Hinz 《manuscripta mathematica》1984,48(1-3):117-137
Let K be an algebraic number field of finite degree over the rationals. The bulk of the present paper is concerned with the problem about the occurrence of algebraic primes in irreducible polynomial sequences generated by polynomials of K with prime arguments from a residue class in K. We shall also deal with the problem of estimating the number of prime idealsT of K withT for which a given integer is a primitive root moduloT, improving on a result of Egami.Dedicated to Professor Dr. W. Schaal on the occasion of his 50th birthday 相似文献
12.
K Györy 《Journal of Number Theory》1982,15(2):164-181
This work is a continuation and extension of our earlier articles on irreducible polynomials. We investigate the irreducibility of polynomials of the form g(f(x)) over an arbitrary but fixed totally real algebraic number field , where g(x) and f(x) are monic polynomials with integer coefficients in , g is irreducible over and its splitting field is a totally imaginary quadratic extension of a totally real number field. A consequence of our main result is as follows. If g is fixed then, apart from certain exceptions f of bounded degree, g(f(x)) is irreducible over for all f having distinct roots in a given totally real number field. 相似文献
13.
It is well known that the Stickelberger–Swan theorem is very important for determining the reducibility of polynomials over a binary field. Using this theorem the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on was determined. We discuss this problem for Type II pentanomials, namely xm+xn+2+xn+1+xn+1F2[x] for even m. Such pentanomials can be used for the efficient implementation of multiplication in finite fields of characteristic two. Based on the computation of the discriminant of these pentanomials with integer coefficients, we will characterize the parity of the number of irreducible factors over F2 and establish necessary conditions for the existence of this kind of irreducible pentanomials.Our results have been obtained in an experimental way by computing a significant number of values with Mathematica and extracting the relevant properties. 相似文献
14.
Optimal extension fields (OEF) are a class of finite fields used to achieve efficient field arithmetic, especially required
by elliptic curve cryptosystems (ECC). In software environment, OEFs are preferable to other methods in performance and memory
requirement. However, the irreducible binomials required by OEFs are quite rare. Sometimes irreducible trinomials are alternative
choices when irreducible binomials do not exist. Unfortunately, trinomials require more operations for field multiplication
and thereby affect the efficiency of OEF. To solve this problem, we propose a new type of irreducible polynomials that are
more abundant and still efficient for field multiplication. The proposed polynomial takes the advantage of polynomial residue
arithmetic to achieve high performance for field multiplication which costs O(m
3/2) operations in
\mathbbFp{\mathbb{F}_p} . Extensive simulation results demonstrate that the proposed polynomials roughly outperform irreducible binomials by 20%
in some finite fields of medium prime characteristic. So this work presents an interesting alternative for OEFs. 相似文献
15.
Robert L. Miller 《Discrete Mathematics》1978,22(1):25-33
The connection between a certain class of necklaces and self-reciprocal polynomials over finite fields is shown. For n?2, self-reciprocal polynomials of degree 2n arising from monic irreducible polynomials of degree n are shown to be either irreducible or the product or two irreducible factors which are necessarily reciprocal polynomials. Using DeBruijn's method we count the number of necklaces in this class and hence obtain a formula for the number of irreducible self-reciprocal polynomials showing that they exist for every even degree. Thus every extension of a finite field of even degree can be obtained by adjoining a root of an irreducible self-reciprocal polynomial. 相似文献
16.
In this paper we study groups for which every real irreducible character has degree not divisible by some given odd prime p. 相似文献
17.
We develop an approach to the character theory of certain classes of finite and profinite groups based on the construction
of a Lie algebra associated to such a group, but without making use of the notion of a polarization which is central to the
classical orbit method. Instead, Kirillov’s character formula becomes the fundamental object of study. Our results are then
used to produce an alternate proof of the orbit method classification of complex irreducible representations of p-groups of nilpotence class < p, where p is a prime, and of continuous complex irreducible representations of uniformly powerful pro-p-groups (with a certain modification for p = 2). As a main application, we give a quick and transparent proof of the p-adic analogue of Brown’s theorem, stating that for a nilpotent Lie group over ℚp the Fell topology on the set of isomorphism classes of its irreducible representations coincides with the quotient topology
on the set of its coadjoint orbits.
The research of M. B. was partially supported by NSF grant DMS-0401164. 相似文献
18.
In a previous work, the authors established various bounds for the dimensions of degree n cohomology and Ext-groups, for irreducible modules of semisimple algebraic groups G (in positive characteristic p) and (Lusztig) quantum groups U ζ (at roots of unity ζ). These bounds depend only on the root system, and not on the characteristic p or the size of the root of unity ζ. This paper investigates the rate of growth of these bounds. Both in the quantum and algebraic group situation, these rates of growth represent new and fundamental invariants attached to the root system ϕ. For quantum groups U ζ with a fixed ϕ, we show the sequence {max L irred dim H n (U ζ , L)} n has polynomial growth independent of ζ. In fact, we provide upper and lower bounds for the polynomial growth rate. Applications of these and related results for are given to Kazhdan-Lusztig polynomials. Polynomial growth in the algebraic group case remains an open question, though it is proved that {log max L irred dim H n (G,L)} has polynomial growth ≤ 3 for any fixed prime p (and ≤ 4 if p is allowed to vary with n). We indicate the relevance of these issues to (additional structure for) the constants proposed in the theory of higher cohomology groups for finite simple groups with irreducible coefficients by Guralnick, Kantor, Kassabov and Lubotzky [13]. 相似文献
19.
We study the explicit factorization of 2
n
r-th cyclotomic polynomials over finite field
\mathbbFq{\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. 相似文献
20.
For non-archimedean local field K and a prime number p we compute the finitely generated pro-p (closed) subgroups of the absolute Galois group of K(t). In addition, we characterize the finitely generated pro-p groups which occur as the maximal pro-p Galois group of algebraic extensions of K(t) containing a primitive pth root of unity. 相似文献