# Isomorphism between finite fields

Refering to this question suppose I have $l(x):=x^3+x+1$ and $m(x):=x^3+x^2+1$. Then prove there is an isomorphism between $\mathbb{F}_3 [x]/l(x)$ and $\mathbb{F}_3[x]/m(x)$

I can say that elements for both the fields are same.

$$x^3+x+1= 0,\quad 1 x,\quad x+1,\quad x^2,\quad x^2+1,\quad x^2+x,\quad x^2+x+1$$

• The polynomials are reducible (both have $1$ as a root). Therefore the quotients aren't fields. – Ayman Hourieh Apr 21 '14 at 18:43
• If you will look at the question that you cite, you will see that the polynomials in question are $x^3-x+1$ and $x^3-x^2+1$, and if you read my answer there, you will see how to construct the isomorphism. And no, you cannot assert that some elements are the same the way you do. – Dilip Sarwate Apr 21 '14 at 18:49
• But I am confused in (a$β^2$+bβ+c)3=a$β^2$+bβ+c−1. How R.H.S is produced? – user120838 Apr 21 '14 at 19:06
• Are you sure you aren't looking at these over $\Bbb{F}_2$? In that case the two polynomials would actually be irreducible. For the actual question Dilip's answer in the linked thread is the way to go. – Jyrki Lahtonen Apr 21 '14 at 20:49

• The polynomials are reducible (both have $1$ as a root). Therefore the quotients aren't fields. – Ayman Hourieh Apr 21 '14 at 18:41