Suppose is finite field with elements, an Galois extension of such that . Prove that is surjective.

From theorem 4.26 we know that is cyclic over generated by .

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is a homomorphism. It is well known that any finite subgroup of the multiplicative group of field is cyclic, which means and is cyclic.

Since and , and , where , then and . Thus is surjective.

Jacobson p300.

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