Degree of A Field Extension - Examples

Examples

• The complex numbers are a field extension over the real numbers with degree = 2, and thus there are no non-trivial fields between them.
• The field extension Q(√2, √3), obtained by adjoining √2 and √3 to the field Q of rational numbers, has degree 4, that is, = 4. The intermediate field Q(√2) has degree 2 over Q; we conclude from the multiplicativity formula that = 4/2 = 2.
• The finite field GF(125) = GF(53) has degree 3 over its subfield GF(5). More generally, if p is a prime and n, m are positive integers with n dividing m, then = m/n.
• The field extension C(T)/C, where C(T) is the field of rational functions over C, has infinite degree (indeed it is a purely transcendental extension). This can be seen by observing that the elements 1, T, T2, etc., are linearly independent over C.
• The field extension C(T2) also has infinite degree over C. However, if we view C(T2) as a subfield of C(T), then in fact = 2. More generally, if X and Y are algebraic curves over a field K, and F : X → Y is a surjective morphism between them of degree d, then the function fields K(X) and K(Y) are both of infinite degree over K, but the degree turns out to be equal to d.