I need help on this problem

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

Thanks

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- Jun 24th 2008, 04:09 PMcapitalXFrobenius map
I need help on this problem

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

Thanks - Jun 24th 2008, 05:04 PMNonCommAlg
i expect someone who's studying algebra knows this much from elementary number theory that for any prime

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

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

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