If p is a prime positive integer, find all subfields of GF(p)

Question

This question just seems too vague.

This has been answered in more generality at http://math.stackexchange.com/questions/91087/subfields-of-finite-fields — Anurag A, Jun 11 '14 at 18:10

Bart Michels · Accepted Answer · 2014-06-11T18:45:59.003

1

Any subfield necessarily contains the multiplicative identity, $1$. Because the characteristic of $\mbox{GF}(p)$ is $p$, we know that $1$ generates an additive subgroup of order $p$, which can only be $\mbox{GF}(p)$ itself. Hence any subfield containing $1$ also contains $\mbox{GF}(p)$. It follows that there are no proper subfields.

edited Jun 11 '14 at 18:45

answered Jun 11 '14 at 18:30

Bart Michels

26,355

Or simply note that $G(p)$ as an additive group has no non-trivial subgroups. – lhf Jun 11 '14 at 19:35

If p is a prime positive integer, find all subfields of GF(p)

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