# Frobenius map

• Jun 24th 2008, 05:09 PM
capitalX
Frobenius map
I need help on this problem

http://i28.tinypic.com/2ryggec.jpg

Thanks
• Jun 24th 2008, 06:04 PM
NonCommAlg
Quote:

Originally Posted by capitalX
I need help on this problem

Let R be a commutative ring of prime characteristic p. Show that the Frobenius map x->x^p is a ring homomorphism from R to R.

Thanks

i expect someone who's studying algebra knows this much from elementary number theory that for any prime

number $p$ and integer $0 < k < p,$ we have: $p \ | \binom{p}{k}.$ so: $\binom{p}{k} \alpha = 0, \ \forall \alpha \in R.$ hence, since R is commutative,

for any $a,b \in R: \ (a+b)^p=a^p+b^p + \sum_{k=1}^{p-1} \binom{p}{k}a^kb^{p-k}=a^p+b^p.$ besides it's obvious that $(ab)^p=a^pb^p.$

therefore the map $x \rightarrow x^p$ is a ring homomorphism. Q.E.D.