Algebraic Extensions

Let F be a field and let E be an extension of F. We say that E is an algebraic extension of F if every E is algebraic over F. One of the main results of this section asserts that if is algebraic over F, then F() is an algebraic extension of F.

Proposition 1: Let E/F be finite, Then E is algebraic over F.

Proof: Let E, n = deg(E/F). The n + 1 elements

1,,...,ⁿ

of E must be linearly dependent over F, since dim_F(E) = n. Therefore, there exist c_i F (0 < i < n), c_i, not all 0, such that

c_nⁿ + c_n-1^n-1 + ... + c₀ = 0.

Therefore, is algebraic over F.