We announce misère-play solutions to several previously-unsolved combinatorial games. The solutions are described in terms of misère quotients—commutative monoids that encode the additive structure of specific misère-play games. We also introduce several advances in the structure theory of misère quotients, including a connection between the combinatorial structure of normal and misère play.

The main purpose of this paper is to define p-adic and q-Dedekind type sums. Using the Volkenborn integral Teichmüller character representations Bernoulli polynomials, we give reciprocity law these These sums their generalized some classical Dedekind law. It be noted that laws, a fine study existing symmetry relations between finite sums, considered in our study, symmetries through permutations...

All rings in this paper are commutative with unity; we will deal mainly with integral domains. Let R be a ring with total quotient ring K. A fractional ideal I of R is invertible if II−1 = R; equivalently, I is a projective module of rank 1 (see, e.g., [Eis95, Section 11.3]). Here, I−1 = (R : I) = {x ∈ K |xI ⊆ R}. Moreover, a projective R-module of rank 1 is isomorphic to an invertible ideal. (...

Let A be a Dedekind domain, K the fraction field of A, and f ∈ A[x] a monic irreducible separable polynomial. For a given non-zero prime ideal p of A we present in this paper a new method to compute a p-integral basis of the extension of K determined by f . Our method is based on the use of simple multipliers that can be constructed with the data that occurs along the flow of the Montes Algorit...

Remark: By the same reasoning, if one replaces 2016 in the problem by a general integer N, then the minimum value of j is the smallest one for which N divides j!. This can be deduced from Pólya’s observation that the set of integer-valued polynomials is the free Zmodule generated by the binomial polynomials (x n ) for n = 0,1, . . . . That statement can be extended to polynomials evaluated on a...

Vaught’s conjecture says that for any countable (complete) first-order theory T, the number of non-isomorphic countable models of T is either countable or 2, where ω is the first infinite cardinal. Vaught’s conjecture for ω-stable theories of modules was proved by Garavaglia [6, Theorem 6]. Buechler proved that Vaught’s conjecture is correct for modules of U-rank 1 [2]. It has been shown that V...

We recall that Theorem 1.3 allows us to define the ideal class group of a Dedekind domain, and in particular of a ring of integers, as the group of fractional ideals modulo the subgroup of principal ideals. We will prove that in the case of a ring of integers, the ideal class group is finite. In fact, we will shortly give a stronger statement due to Minkowski. Using similar techniques, we will ...

In this paper we provide a constructive version of Tits alternative for a broad class of quaternions with algebraic coefficients. Our result is a generalization of that contained in the paper [1], concerning groups of rational quaternions. Indeed, the tools developed in [1] can be extended to arbitrary number fields by translating them in the corresponding Dedekind domain, as the techniques inv...

These notes record the basic results about DVR’s (discrete valuation rings) and Dedekind rings, with at least sketches of the non-trivial proofs, none of which are hard. This is standard material that any educated mathematician with even a mild interest in number theory should know. It has often slipped through the cracks of Chicago’s first year graduate program, but then we would need at least...