# Finding exact isomorphism between finite fields given as quotient rings [duplicate]

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I have two quotient rings over $\Bbb F := GF(3)$: $$\Bbb F[x] / (x^3 -x - 1) \qquad \text{and} \qquad \Bbb F[x] / (x^3 -x + 1) .$$

These things I know: Both quotient rings are irreducible, that means they are the same size, so there should exist isomorphism.

I think these two quotient rings should be a field because they have maximum ideals.

I am trying to find function phi to create isomorphism between these quotient rings but not using brute force method. I have tried several ways but they are not "easy" enough. I know that identity element must be mapped to identity element but it didn't help me to find that exact isomorphic function.

Can somebody push me the correct way or give me tip how and where should I look to easily find isomorphism between these quotient rings without necessity to "heavy" computing.

## marked as duplicate by Jyrki Lahtonen abstract-algebra StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Apr 24 '16 at 17:49

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• +1s to you all. But this has been asked before. – Jyrki Lahtonen Apr 24 '16 at 17:49

## 1 Answer

Hint $$(-x)^3 - (-x) - 1 = -(x^3 - x + 1) .$$

• I dont see the connection yet so i can build that isomorphism :/ Could you please specify it a bit more ? – user3411282 Apr 24 '16 at 16:09
• In the variable $y := -x$, $y^3 - y - 1 = -(x^3 - x + 1)$, and (introducing a different name for the dummy indeterminate in one case for emphasis), we want to find an isomorphism between $\Bbb F_3[y] / \langle y^3 - y - 1 \rangle$ and $\Bbb F_3[x] / \langle x^3 - x + 1 \rangle$. – Travis Apr 24 '16 at 16:15