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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 alpha element of E is algebraic over F. One of the main results of this section asserts that if alpha is algebraic over F, then F(alpha) is an algebraic extension of F.

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

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

1,alpha,...,alphan

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

cnalphan + cn-1alphan-1 + ... + c0 = 0.

Therefore, alpha is algebraic over F.



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