Abstract Algebra: Theory and Applications · Problem 22.4.18

Galois Theory — Thomas Judson

Mathematics — Abstract Algebra · finite-fields-degree-extensions

Problem

Let EE be an extension of a finite field FF, where FF has qq elements. Let α∈E\alpha \in E be algebraic over FF of degree nn. Prove that F(α)F(\alpha) has qnq^n elements. Hint. Since α\alpha is algebraic over FF of degree nn, we can write any element β∈F(α)\beta \in F(\alpha) uniquely as β=a0+a1α+⋯+an−1αn−1\beta = a_0 + a_1\alpha + \cdots + a_{n-1}\alpha^{n-1} with ai∈Fa_i \in F. There are qnq^n possible nn-tuples (a0,a1,…,an−1)(a_0, a_1, \ldots, a_{n-1}).

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