Kurt Girstmair

On the computation of resolvents and Galois groups

A large class of algorithms for computing resolvents of algebraic equations — so called rational transformations — is investigated and characterized group theoretically. The concept of rational transformations implies a program how to develop good methods to determine the Galois group of an equation. It is shown that some known methods are special cases of rational transformations, and a new procedure to find the group of a sextic equation is given. Moreover, all cases in which Galois resolvents can be found by means of rational transformations are classified.

A “Popular” Class Number Formula

we get x1 + X3 + * * * +X45 = 76 and X2 + X4 + * * * +X46 = 131, whose difference is 55. Hence the average of x1, X3, . . . iS 323, whereas the average Of X27 X4, . . . iS c 16 J23 The Theorem holds for the digit expansion of 1/p with respect to an arbitrary basis g E @, g > 2, provided that g is a primitive root mod p. The property of being primitive can be characterized as follows: For k = 0,1,2,3,. . . define gk E {0, 1, . . . , p 1} by

An index formula for the relative class number of an abelian number field

Abstract Let n be the conductor of an imaginary abelian number field K, O the ring of algebraic integers of K, and Q n the nth cyclotomic field. We describe the index of the additive group generated by the conjugate elements of the trace Tr Q n/K(i·cot(π/n)) in the group O ⌣ i· R (if n = pr is a prime power, one has to take Tr Q n/K(ip·cot(π/n)) instead). This index equals the relative class number of K, multiplied with a factor that is explicitly given in terms of the ramification of K over Q . In some respect this result is an analogue of the representation of the class number of K ⌣ R as an index of circular units, rather than the hitherto known Stickelberger index formulas of Iwasawa et al. We also show that the higher derivatives i m cot (m − 1) ( π n ) , m ≥ 2, yield index formulas analogous to the higher Stickelberger formulas of Kubert and Lang.

Character coordinates and annihilators of cyclotomic numbers

The object of this paper is a representation theoretical approach to the problem of determining allQ-linear relations between conjugate numbers in a cyclotomic field. We apply our method to relations between the numbers cot(m)(πk/n), tan(m)(πk/n), cosec(m)(2πk/n), sec(m)(2πk/n), respectively, where m is≥0 and (k,n)=1. Thereby we complete previous work of Chowla, Hasse, Jager-Lenstra, and others.

A CRITERION FOR THE EQUALITY OF DEDEKIND SUMS MOD ℤ

In [Jabuka, Robins and Wang, When are two Dedekind sums equal? Internat. J. Number Theory7 (2011) 2197–2202], it was shown that the Dedekind sums s(m1, n) and s(m2, n) are equal only if (m1m2-1)(m1-m2) ≡ 0 mod n. Here we show that the latter condition is equivalent to 12s(m1, n) - 12s(m2, n) ∈ ℤ. In addition, we determine, for a given number m1, the number of integers m2 in the range 0 ≤ m2 < n, (m1, m2) = 1, such that 12s(m1, n) - 12s(m2, n) ∈ ℤ, provided that n is square-free.

Class number factors and distribution of residues

LetGm/H be a factor group of the prime residue groupGm = (ℤ/mℤ)x. We introduce theb-division vectorT(b), which contains important data on the distribution of the integersk, 1 ≤k ≤m, (k, m) = 1, amongst the classesC ε Gm/H. Of particuar interest are the euclideanlength ofT(b) and thesigns of its components. First we express certain class number factors of abelian subfields of them-th cyclotomic field in terms ofT(b). These expressions can be used as class number formulas, but we are more interested in the converse question: What is the influence of the said class number factors onT(b), in particular, on ‖T(b)‖? Much information about this influence is contained in thenorm manifold ℒ. This manifold is a family of euclidean tori, which we study in detail. In some cases ℒ. is compact, which means that class number factors determine the length ofT(b) completely. In the remaining cases ℒ. supplies a lower bound for ‖T(b)‖ only. We obtain some insight into the quality of this lower bound by means of the Generalized Riemann Hypothesis. Moreover, we show that the connection ofT(b) with class number factors yields interesting facts about the signs of the components ofT(b) and is helpful in actual computations.

On the fractional parts of Dedekind sums

We show that each rational number r, 0 ≤ r < 1, occurs as the fractional part of a Dedekind sum S(m, n). Further, we determine the number of integers x, 1 ≤ x ≤ n, (x, n) = 1, such that S(m, n) and S(x, n) have the same fractional parts.

Dirichlet convolution of cotangent numbers and relative class number formulas

Letn be the conductor of an abelian number fieldK. The numbersicot (πk/n), (k, n)=1, belong to, then-th cyclotomic field; theirK-traces form an additive group whose index in the “imaginary part” of the ringOK involves the relative class numberhK− ofK. This was shown previously. In the present paperhK− is decomposed into “branch factors”, each of which is shown to be the index of an additive group of modified cotangent numbers. Put together in the right way, the said numbers yield formulas forhK− simpler than the previous ones. The different types of cotangent numbers are mutually connected by Dirichlet convolution, whose meaning in the construction of cyclotomic numbers is studied. Finally, our results are rephrased in terms of Stickelberger ideals.

CONTINUED FRACTIONS AND JACOBI SYMBOLS

We consider the regular continued fraction expansion of a rational number m/N, m ≥ 0, N ≥ 1, (m, N) = 1. Let s/t, (s, t) = 1, be the kth convergent of this expansion and p/q, (p, q) = 1, be the complete quotient belonging to s/t. We give some relations for Jacobi symbols, a typical example of which is for k, t, q, N odd, with a simple right-hand side depending on t, q, N (mod 4). As an application, we prove the periodicity of the Jacobi symbol for the convergents s/t of infinite purely periodic continued fractions.

Über konstruktive Methoden der Galoistheorie.

The Definition of the solution of an algebraic equation introduced in this note involves the explicite construction of the splitting-field and the exhibition of the roots in this field. It is shown in which cases a solution may be found by means of rational functions only. Furthermore a feasible solution-method is presented. This method is used to tackle other problems of Galois theory, too. As an example, the factorization of polynomials can be reduced to the determination of zeros of polynomials.