How many finite fields with 10 elements are there?

Question

I want to know how many finite fields with 10 elements there are. I know for example $\mathbb Z_{11}$ is one of them. How can I count all the possible polynomial fields or even something like $\mathbb Z^{*}_{p}$? Or is $\mathbb Z_{11}$ the only one?

thank you for your help

The number of elements of a finite field must be a prime power, so there is no field with $10$ elements — Peter, Jul 09 '17 at 14:32
A field $(P,+,*)$ itself is an abelian group for $(P,+)$, so if $P$ is non-trivial, then it has to be a $p$-field for some prime $p$. — user771160, Jul 09 '17 at 14:37
For every prime number $p$ and every positive integer $n$, there exist finite fields of order $p^n$, and all fields of this order are isomorphic. — Ben Grossmann, Jul 09 '17 at 14:38
@Peter Yes of course. I have deleted. Actually I meant $n\geq1$ (true prime power). — drhab, Jul 09 '17 at 15:14

score 10 · Accepted Answer · answered Jul 09 '17 at 14:36

10

None. A finite field $F$ always has a prime subfield $\mathbf F_p=\mathbf Z/p\mathbf Z$ for some prime $p$ ($p$ is the smallest natural number $n$ such that $n\cdot 1=0$) and it is a finite dimensional vector space over this prime field, hence if its dimension is $r$, it has $p^r$ elements. As a consequence, the cardinality of a finite field is some power of a prime number.

answered Jul 09 '17 at 14:36

Bernard

175,478

Yeah you are right, it is $\mathbb Z \to F$. I was preoccupied just now. – user771160 Jul 09 '17 at 15:01

How many finite fields with 10 elements are there?

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