Proof that a factor ring is a field

Asked Apr 01 '22 at 20:38

Active Apr 01 '22 at 20:38

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Let $f(x)=x^3+x^2+1\in\mathbb{Z}_2[x]$, and $I=f(x)\mathbb{Z}_2[x]$. Show that $\mathbb{Z}_2[x]/I$ is a field and that it contains $8$ elements.

I think it is related to the fundamental theorem of ring homomorphisms, which says $R/Ker f\cong Im f $; we know the image is $\mathbb{Z}_2[x]$. In other words, if $Ker f=I$ then we are done. I fail to see how this could be the case. I have no clue how to prove this result though. I can explain why there are eight elements though ; if a polynomial of degree $p$ is in $\mathbb{Z}_q$ then there are $q^p$ elements, so $2^3=8$ in our case. It is the first part I am unsure about.

asked Apr 01 '22 at 20:38

You could perhaps show that the quotient ring has no zero-divisors. – Hagen von Eitzen Apr 01 '22 at 20:45
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If $A$ is a ring and $I\subset A$ an ideal, then $A/I$ is a field iff $I$ is maximal. – Douglas Molin Apr 01 '22 at 20:56
You can always prove this directly by using polynomial long division to characterize the elements of the quotient space. – CyclotomicField Apr 01 '22 at 21:20

Proof that a factor ring is a field

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