Gauss’ lemma and valuation theory

Fundamental Properties of Valuation Theory and Hensel's Lemma

Convergence with respect to a valuation is discussed as convergence of a Cauchy sequence. Cauchy sequences of polynomials are defined. They are used to formalize Hensel’s lemma.

Gauss Lemma and Law of Quadratic Reciprocity

The papers [20], [10], [9], [11], [4], [1], [2], [17], [8], [19], [7], [16], [13], [21], [22], [5], [18], [3], [15], [6], and [23] provide the terminology and notation for this paper. For simplicity, we adopt the following convention: i, i1, i2, i3, j, a, b, x denote integers, d, e, n denote natural numbers, f , f ′ denote finite sequences of elements of Z, g, g1, g2 denote finite sequences of ...

Regular hypergraphs, Gordon's lemma, Steinitz' lemma and invariant theory

Let D(n)(D(n, k)) denote the maximum possible d such that there exists a d-regular hypergraph (d-regular k-uniform hypergraph, respectively) on n vertices containing no proper regular spanning subhypergraph. The problem of estimating D(n) arises in Game Theory and Huckemann and Jurkat were the first to prove that it is finite. Here we give two new simple proofs that D(n), D(n, k) are finite, an...

Valuation theory of exponential

Valuation Theory. Part I

In the article we introduce a valuation function over a field [1]. Ring of non negative elements and its ideal of positive elements have been also defined. The notation and terminology used here have been introduced in the following We use the following convention: x, y, z, s are extended real numbers, i is an integer, and n, m are natural numbers. The following propositions are true: (1) If x ...