Galois field

Finite extension of a Galois field

Let be a prime and be integers. Then there exists an extension

iff . Moreover, when such an extension exists it is unique and simple.¹ field

Proof

Suppose such an extension exists, whence we have a tower of field extensions , so in particular divides , i.e. .

Conversely, assume , whence , and by the same token . Therefore . By Construction and uniqueness, is the splitting field of the second polynomial. It follows from ^P1 that .

For the last statement, note that Finite subgroup of the group of units of a field is cyclic, so in particular has a generator , which will necessarily generate over any subfield. If this means , so the extension is simple.

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Footnotes

1. 2009. Algebra: Chapter 0, §VII.5.1, p. 442. ↩