Christian U. Jensen

Polynomials with Dp as Galois group

Abstract A useful criterion characterizing a monic irreducible polynomial over Q 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.

Homological Dimension and representation type of algebras under base field extension.

We prove that for finite-dimensional associative algebras the finitistic global dimension in the sense of H. Bass and also the self-injective dimension are preserved under arbitrary extensions of the base field. Further, the global dimension, and as a consequence also finite representation type, are preserved under base field extensions which are separable in the sense of S. MacLane. The results are derived with the aid of ultraproduct-techniques and are applied to the model theory of finite dimensional algebras.

Polynomials with Frobenius groups of prime degree as Galois groups II

Abstract We present characterization theorems for polynomials of prime degree p ≥ 5 over Q with Frobenius groups of degree p: Fpl = Flp′, l| p − 1 as Galois groups. We construct a generic family of polynomials with Galois group F p(p − 1) 2 (p ≡ 3 (mod 4)), using the pth Chebyshev polynomial of the first kind. In addition, a parametric family of quintic polynomials with Galois group F20 is given. Explicit examples of polynomials are presented for F2p = Dp (p ≤ 19), F20 and for F p(p − 1) 2 with p = 7 and 11. Some properties of Frobenius fields are also discussed.

Some curiosities of rings of analytic functions

1. It is the purpose of this paper to exhibit some properties of certain rings of analytic functions which may be a little unexpected. Let E be the ring of all entire functions in one complex variable, i.e. the subring of C[[X]] consisting of all formal power series with infinite convergence radius. More generally, for a subfield K of C let E(K) be the subring of K[[X]] formed by all power series with infinite convergence radius. If Q is a positive real number, let E(Q, K) , resp. J?(Q, K) be the subring of K[[X]] consisting of all power series with convergence radius >Q, resp. 2~. It is well known that E(K) is a non-Noetherian domain and that E(K) is a Bezout domain, i.e. any finitely generated ideal is principal. If K=C this was proved by Wedderburn [ 131 and in the general case by Helmer [4], (who apparently was unaware of [13]). As for the Krull dimension of E the first ‘result’ appeared in [lo] stating that K-dim E = 1. An error in the proof was noticed by Kaplansky [5] and K-dim E is actually infinite. We shall give more precise results concerning the length of chains of prime ideals of E. As shown in [7] the global dimension of E is 23, while the exact value of gl.dimE cannot be determined from the usual axioms of set theory (ZFC): For any t, 3 5 tr m, the statement gl.dim E = t is consistent with ZFC, in fact, even consistent with ZFC + MA, (MA denoting Martin’s axiom). The corresponding results hold true if E is replaced by E(K) or &, K). The proofs only require minor modifications. For E(Q, K), however, the situation is completely different. For any positive Q and any field KS C the ring E(Q, K) is Euclidean, in particular a PID. Since for instance E(1, C) = n;=, E(l l/n, 4X) we obtain a decreasing sequence of PID’s whose intersection is a Bezout domain of undecidable global dimension and of uncountable Krull dimension. The stable range (in the sense of Bass [l]) of the above rings depends on K. If KS R the stable range of each of the rings E(Q, K), E(Q, K) and E(K) is 2, otherwise, when Kg R the stable range is 1.

Some cardinality questions for flat modules and coherence

All rings considered in this paper will be associative with an identity element and all modules will be unitary. In Section 1 we shall consider some questions concerning the representation of a flat module as a directed union of flat submodules with a prescribed number of generators, in particular give a partial solution of the problem of determining those rings for which any flat module is a directed union of finitely generated projective submodules. In Section 2 we shall consider some questions about the weak dimension of power series rings over von Neumann regular rings.

A remark on the distributive law for an ideal in a commutative ring

Let R be a commutative ring, with an identity element. It is the purpose of this note to establish conditions for an arbitrary but fixed ideal a of R to satisfy the distributive law for all ideals b and c of R . In particular, in the Noetherian case, this will be related to the decomposition of a into prime ideals. We start with Proposition 1. For a fixed ideal a in a commutative ring R with an identity element, the following conditions are equivalent .

On the axiomatizability of certain classes of modules

0. Let R be a fixed associative ring with an identity element. Unless the contrary is stated explicitly, by a module we mean a left R-module. In the present paper we consider the first order language (which allows only finite conjunctions and disjunctions) whose only non-logical constants are the equality symbol, a constant 0 and the following function symbols: a binary function f and for each r~R, a unary function grWe write x + y for f ( x , y ) and r . x for gr(x). In the obvious way any R-module becomes a structure for this language, which we call the language of (left) R-modules and denote by L(R). We recall the following basic notions from model theory. A class cg of R-modules is called elementary in the wider sense or axiomatizable (EC~ in the notation of [1]) if cg is the class of models of a set of first order sentences in L(R). A class cg of R-modules is called elementary in the strict sense or finitely axiomatizable (EC in the notation of [13) if ~ is the class of models of a single first order sentence in L(R). A class ~ of R-modules is called elementarily closed (EC~s in the notation of [1]) if any R-module elementarily equivalent to an R-module in ~ belongs to c~. A basic result concerning the above classes is (cf. [1, 3]):

Centres and fixed-point rings of artinian rings

It is well known that the centre of a left artinian ring R is usually not artinian; however considering the centre Z(R) as the endomorphism ring of the (R | R~ R of finite length, it follows that Z(R) is necessarily semiprimary, i.e. the Jacobson radical J(R) is nilpotent and R/J(R) is artinian. It is the purpose of this note to describe (partially) the commutative semiprimary rings which can appear as centres of artinian rings. In Section 1 we show that if a commutative ring R is the fixed-point ring for a finitely generated group of automorphisms of an artinian ring, then R is also centre of an artinian ring. In Section 2 we firstly prove that any commutative semiprimary ring, for which the square of the Jacobson radical vanishes, can be embedded in an artinian ring. Next we see that many of these semiprimary rings actually appear as centres of artinian rings. Here the assumption that the square of the Jacobson radical is zero, is essential as shown by Example 2.1. In this paper a ring will always mean an associative ring with an identity element, subrings have the same identity element and homomorphisms preserve the identity element. J(R) denotes the Jacobson radical of the ring R.

Embeddability of quadratic extensions in cyclic extensions

Abstract For an algebraic number field K we study the quadratic extensions of K which can be embedded in a cyclic extension of K of degree 2 n for all natural numbers n, as well as the quadratic extensions which can be embedded in an infinite normal extension with the additive group of 2-adic integers as Galois group. For shortness we call a normal extension of K whose Galois group is the cyclic group ℤ/2 n ℤ of order 2 n with n ∈ ℕ, resp. , a (ℤ/2 n ℤ)-extension resp. a -extension of K. A quadratic extension L|K is called (ℤ/2 n ℤ)-embeddable, resp. -embeddable, if there exists a (ℤ/2 n ℤ)-extension, resp. a -extension, of K containing L. One main result of this paper is the following observation, the exact formulation of which is given in theorems 6 to 8 in §3: Theorem 0. Let K be an imaginary quadratic number field whose discriminant has m prime divisors. Then the number of quadratic extensions L|K which are (ℤ/2 n ℤ)-embeddable for all n is 2 m−1 − 1, 2 m − 1 or 2 m+1 − 1, depending on certain congruences for the discriminant and its prime divisors. But the number of quadratic extensions L|K which are -embeddable is only 3.

Unique realizability of finite abelian 2-groups as Galois groups

Let n ≥ 1 and μi ≥ 0 for 1 ≤ i ≤ n. We prove that to each group G = Z2ν1μ1 × ⋯ × Z2νnμn with 1 = ν1 < ⋯ < νn and n − 2 ≤ μn there exists a field K of characteristic zero such that G can be realized as Galois group Gal(NK) of exactly one normal extension N of K. We also show that K can be chosen algebraic over Q if and only if μ2 + ⋯ + μn − 1 ≤ μn. Moreover, we give a complete description of the finitely generated pro-2-groups which occur as Galois groups of maximal abelian 2-extensions of (infinite) algebraic number fields.