2 Questions: Fields and Subfields

**GB89** · February 12th 2009, 07:28 PM

1.) F is a field of prime characteristic p. Prove that K = {x in F for all x^p=x} is a subfield of F.

For the subfield test, I showed that a-b is in K whenever a and b are in K, but I'm getting stuck when I try to show that ab inverse is in K.

2.) Show that any finite field has order p^n, where p is prime.

I have that Char F = p and the order of a divides p, and I know that the order of a is either 1 or p. I also know that p divides the order of the field, so F must contain an element of order p. I'm just having trouble connecting that to proving any finite field has order p^n.

Any help would be greatly appreciated. Thanks.

**NonCommAlg** · February 12th 2009, 08:13 PM

Originally Posted by GB89

1.) F is a field of prime characteristic p. Prove that K = {x in F for all x^p=x} is a subfield of F.

For the subfield test, I showed that a-b is in K whenever a and b are in K, but I'm getting stuck when I try to show that ab inverse is in K.

continuing what you've already done: 0 and 1 are in K. also if $0 \neq a \in K,$ then $(a^{-1})^p=(a^p)^{-1}=a^{-1}.$ so $a^{-1} \in K.$ also if $a,b \in K,$ then $(ab)^p=a^pb^p=ab.$ so $ab \in K.$

2.) Show that any finite field has order p^n, where p is prime.

I have that Char F = p and the order of a divides p, and I know that the order of a is either 1 or p. I also know that p divides the order of the field, so F must contain an element of order p. I'm just having trouble connecting that to proving any finite field has order p^n.

Any help would be greatly appreciated. Thanks.

$f: \mathbb{Z} \longrightarrow F$ defined by $f(n)=n1_F$ is a ring homomorphism and $\ker f = p \mathbb{Z}.$ thus $f$ contains a field $\mathbb{F}_p \simeq \frac{\mathbb{Z}}{p\mathbb{Z}}.$ hence $F$ can be considered as a finite dimensional vector space over $\mathbb{F}_p.$

let $[F:\mathbb{F}_p]=n.$ then $|F|=p^n$ because $|\mathbb{F}_p|=p.$

**ThePerfectHacker** · February 12th 2009, 08:19 PM

Originally Posted by GB89

1.) F is a field of prime characteristic p. Prove that K = {x in F for all x^p=x} is a subfield of F.

Hint: $(a+b)^p = a^p + b^p \text{ and }(ab)^p = a^pb^p$ .

2.) Show that any finite field has order p^n, where p is prime.

Define $\phi:\mathbb{Z}\to F$ by $\phi(n) = n\cdot 1$ where $1$ is multicative identity element of $F$ .
Prove that $\phi$ is a ring homomorphism and $\ker(\phi) = p\mathbb{Z}$ since $\text{char}(F) = p$ .
It means that $\mathbb{Z}_p\simeq \phi[ \mathbb{Z}]\subseteq F$ .
Therefore, $F$ contains a subfield of order $p$ .
Since $F$ is a vector space over this smaller field forces $|F| = p^n$ for some $n$ .

EDIT: Too slow.

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