# Advanced Algebra:commutative ring

Let x and y belong to a commutative ring R with prime characteristic p.

a) Show that (x + y)^p = x^p + y^p

b) Show that, for all positive integers n, (x + y)^p^n = x^p^n + y^p^n.

c) Find elements x and y in a ring of characteristic 4 such that (x + y)^4 != x^4 + y^4.

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#### Solution Preview

We know the formula nCk= n! / k! (n-k)!

a) (x+y)^p = x^p + pC1 x^p-1 y + pC2 x^p-2 y^2 +... + y^p

In a commutative ring with prime characteristic p all multiples of p are zero

Since all ...

#### Solution Summary

A commutative ring is contextualized.

$2.19