Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits, i.e., decomposes into linear factors. Definition A splitting field of a polynomial p(X) over a field K is a field extension L of K over which p factors into linear factors :p(X) = c\prod_{i=1}^{\deg(p)} (X - a_i) where c\in K and for each i we have X - a_i \in L[X] with ai not necessarily distinct and such that the roots ai generate L over K. The extension L is then an extension of minimal degree over K in which p splits. It can be shown that such splitting fields exist and are unique up to isomorphism. The amount of freedom in that isomorphism is known as the Galois group of p (if we assume it is separable). Properties An extension L which is a splitting field for a set of polynomials p(X) over K is called a normal extension of K. Given an algebrai

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MATH-310: Algebra

Study basic concepts of modern algebra: groups, rings, fields.

Splitting fields of conics and sums of squares of rational functions

David Maximilian Grimm

Given a geometrically unirational variety over an infinite base field, we show that every finite separable extension of the base field that splits the variety is the residue field of a closed point. As an application, we obtain a characterization of function fields of smooth conics in which every sum of squares is a sum of two squares.

Springer-Verlag2013

Actions de relations d'équivalence sur les champs d'espaces métriques CAT(0)

Philippe Paul Antoine Henry

This work is dedicated to the study of Borel equivalence relations acting on Borel fields of CAT(0) metric spaces over a standard probability space. In this new framework we get similar results to some theorems proved recently by S. Adams-W. Ballmann or N. Monod concerning groups of isometries of CAT(0) spaces. In Chapter 1, we build several Borel structures on a variety of fields before dealing in particular with Borel fields of CAT(0) spaces. Chapter 2 discusses the notion of an action for an equivalence relation on a field of metric spaces and gives several examples. We also introduce a definition of amenability for equivalence relations in terms of invariant section following an idea of R.J. Zimmer. Chapter 3 deals with the action of an amenable equivalence relation and shows that such a relation cannot act without fixing a section at infinity or preserving a subfield of Euclidean spaces. In Chapter 4, we show that if an equivalence relation is generated by two commuting groups and acts without fixing a section at infinity, then the field splits equivariantly and isometrically as a product. Using this result we also show that equivalence relations containing two coamenable subrelations cannot act without fixing a section at infinity or preserving a subfield of Euclidean spaces.

EPFL2010

Computing irreducible representations of supersolvable groups over small finite fields

Mohammad Amin Shokrollahi

We present an algorithm to compute a full set of irreducible representations of a supersolvable group G over a finite field K, charK |G|, which is not assumed to be a splitting field of G. The main subroutines of our algorithm are a modification of the algorithm of Baum and Clausen (Math. Comp. 63 (1994), 351-359) to obtain information on algebraically conjugate representations, and an effective version of Speiser's generalization of Hilbert's Theorem 90 stating that H[1](Gal(L/K), GL(n, L)) vanishes for all n >= 1

American Mathematical Society1997

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus

Field extension

In mathematics, particularly in algebra, a field extension is a pair of fields K\subseteq L, such that the operations of K are those of L restricted to K. In this case, L is an extension

Group (mathematics)

In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inver