Separable extensionIn field theory, a branch of algebra, an algebraic field extension is called a separable extension if for every , the minimal polynomial of over F is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field). There is also a more general definition that applies when E is not necessarily algebraic over F. An extension that is not separable is said to be inseparable.
Galois extensionIn mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.
Glossary of field theoryField theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) A field is a commutative ring (F,+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division. The non-zero elements of a field F form an abelian group under multiplication; this group is typically denoted by F×; The ring of polynomials in the variable x with coefficients in F is denoted by F[x].
Splitting fieldIn 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. 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 where and for each we have 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.
Frobenius endomorphismIn commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its p-th power. In certain contexts it is an automorphism, but this is not true in general. Let R be a commutative ring with prime characteristic p (an integral domain of positive characteristic always has prime characteristic, for example).
Field extensionIn mathematics, particularly in algebra, a field extension is a pair of fields such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.
