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The extension field K of a field F is called a splitting field for the polynomial f(x) in F[x] if f(x) factors completely into linear factors in K[x] and f(x) does not factor ...

A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra. An archaic name for a field is ...

Given a set P of primes, a field K is called a class field if it is a maximal normal extension of the rationals which splits all of the primes in P, and if P is the maximal ...

Take K a number field and m a divisor of K. A congruence subgroup H is defined as a subgroup of the group of all fractional ideals relative prime to m (I_K^m) that contains ...

The following are equivalent definitions for a Galois extension field (also simply known as a Galois extension) K of F. 1. K is the splitting field for a collection of ...

A field K is said to be an extension field (or field extension, or extension), denoted K/F, of a field F if F is a subfield of K. For example, the complex numbers are an ...

A prime field is a finite field GF(p) for p is prime.

A perfect field is a field F such that every algebraic extension is separable. Any field in field characteristic zero, such as the rationals or the p-adics, or any finite ...

A polynomial p(x)=sumc_ix^i is said to split over a field K if p(x)=aproduct_(i)(x-alpha_i) where a and alpha_i are in K. Then the polynomial is said to split into linear ...

A field F in which any Pythagorean extension of F coincides with F.