Are $R_1=\mathbb{F}_5[x]/(x^2+2)$ and $R_2=\mathbb{F}_5[y]/(y^2+y+1)$ isomorphic rings? If so, write down an explicit isomorphism. If not, prove they are not.

My Try:

Since $x^2+2$ is irreducible in $\mathbb{F}_5[x]$, and $y^2+y+1$ is irreducible in $\mathbb{F}_5[y]$, both $R_1$ and $R_2$ are fields. Moreover, $O(R_1)=25=O(R_2)$. Since for a given prime $p$ and integer $n$ there is a unique field with $p^n$ elements, $R_1$ and $R_2$ are isomorphic. But how can I write an explicit isomorphism? Can somebody please help me to find it?

  • 1
    $\begingroup$ Did you try the obvious one that sends $ax+b+(x^2+2)$ to $ay+b+(y^2+y+1)$, and see if that worked? (I'm not sure if it will, but that would be the first try) $\endgroup$ – Alan Aug 9 '15 at 22:57
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    $\begingroup$ @Alan That won't work, since $x\mapsto y$ will imply that $0=x^2+2 \mapsto y^2+2=-y-1\neq 0$. However, $x\mapsto y+3$ will work wonderfully. $\endgroup$ – Batominovski Aug 9 '15 at 23:17

Hint: $(y + 3)^2 + 2 = y^2 + y + 1$ as elements of $\mathbb F_5[y]$.


The start of your argument is fine. For the explicit isomorphism note that $x$ is a square root of $-2$ in the first, now find one in the second too.

To this end note that $y^2 + y +1 = (y^2+ y -1) +2 = (y^2+ 4y +4) +2 = (y+2)^2 +2 $.


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