JaneBennet
04-20-2008, 08:05 PM
If E is a field and F is a subset of E containing 0 and 1 such that F is closed under addition and multiplication and the additive and multiplicative inverse of every element in F is in F (except that 0 does not have a multiplicative inverse), then F is said to be a subfield of E. Equivalently, E is called an extension field of F.
If E is an extension field of F, then E can be considered as a vector space over F. The dimension of this vector space is called the degree of the field extension of E over F.
The set of all automorphisms of the field E which fix every element in the subfield F is a subgroup of the automorphism group of E. Such a subgroup has an important part to play in Galois theory.
If E is an extension field of F, then E can be considered as a vector space over F. The dimension of this vector space is called the degree of the field extension of E over F.
The set of all automorphisms of the field E which fix every element in the subfield F is a subgroup of the automorphism group of E. Such a subgroup has an important part to play in Galois theory.