# Subring of a Field with characteristic P, a prime number.

Suppose $F$ is a field with characteristic $p$, a prime number. Prove that $F$ contains a subring identical to $Z_p$.

Are identical subrings the same? There is no mention of this in the text. So do I only have to prove that $Z_p$ is a subring of the defined field?

Then if $x,y$ are elements of $F$, then $(x+y)$ is an element of $F$. If $(x+y) < p$, then its also an element of $Z_p$. If $(x+y)=mp$, where $m$ is any natural number, then $(x+y)=0$ and is still closed under addition in $Z_p$, but couldn't $(x+y)>p$, and thus it wouldn't be closed in $Z_p$ or am I conceptualizing this wrong some how (ring theory is a really new concept).

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Identical means isomorphic – Amr Nov 9 '12 at 1:37
I know nothing about isomorphic rings...how does one show they are isomorphic? – Carly Nov 9 '12 at 1:45
By showing that there exists a bijection that preserves the operations – Amr Nov 9 '12 at 1:47

Consider the subring $R=${${0,1,1+1,...}$}