I don't really understand polynomial quotient rings

**electricjaguar** · March 17th 2009, 07:35 PM

For example, what is the difference between

$\mathbb{Z}_2[x]/(x^2 + 1)$ and $\mathbb{Z}_2[x]/(x^2 + x + 1)$ ?
What do they look like?

Is $\mathbb{Z}_2[x]/(x^2 + 1) \cong \mathbb{F}_2[x]/(x^2 + 1)$ ?

I'm having trouble "getting it" if that makes sense.

**GaloisTheory1** · March 17th 2009, 07:52 PM

Originally Posted by electricjaguar

For example, what is the difference between

$\mathbb{Z}_2[x]/(x^2 + 1)$ and $\mathbb{Z}_2[x]/(x^2 + x + 1)$ ?
What do they look like?

Is $\mathbb{Z}_2[x]/(x^2 + 1) \cong \mathbb{F}_2[x]/(x^2 + 1)$ ?

I'm having trouble "getting it" if that makes sense.

$\mathbb{Z}_2[x]/(x^2 + 1)$ $= \{ a+bx+x^2 + 1|a,b \in \mathbb{Z}_2 \}$

$\mathbb{Z}_2[x]/(x^2 + x + 1)$ = $\{ a+bx+x^2 +x+ 1|a,b \in \mathbb{Z}_2 \}$

make sure the polynomials are irreducible. also, the fields have the same cardinality [which is 4].

Yes b/c $F_2[x]=\mathbb{Z}/2\mathbb{Z} \cong \mathbb{Z}_2$

**electricjaguar** · March 17th 2009, 10:08 PM

Thank you, I have more questions that I'm just going to put in this thread so as not to cloud up the forum.

Is it true that $\mathbb{Z}/(16)$ contains a unique maximal ideal, but $\mathbb{Z}/(12)$ does not? I think this is the case by the correspondence theorem

Also, I can see why $\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}$ , but why is $\mathbb{R}[x]/(x^2 + x + 1) \cong\mathbb{C}$ ?

**electricjaguar** · March 18th 2009, 04:01 AM

I also have a question about $\mathbb{Z}_2[x]/(x^2 + 1)$ . Isn't one of its elements $x^2 + 1 + 1 = x^2 + 2 = x^2$ which has a root in $\mathbb{Z}_2$ , namely, x = 0? Doesn't this show that this element is reducible and hence the quotient is not a field?

edit: nevermind

**GaloisTheory1** · March 18th 2009, 08:47 AM

Originally Posted by electricjaguar

Thank you, I have more questions that I'm just going to put in this thread so as not to cloud up the forum.

Is it true that $\mathbb{Z}/(16)$ contains a unique maximal ideal, but $\mathbb{Z}/(12)$ does not? I think this is the case by the correspondence theorem

Also, I can see why $\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}$ , but why is $\mathbb{R}[x]/(x^2 + x + 1) \cong\mathbb{C}$ ?

yes b/c $8\mathbb{Z}/16\mathbb{Z} \subseteq 4\mathbb{Z}/16\mathbb{Z} \subseteq 2\mathbb{Z}/16\mathbb{Z}$
so $2\mathbb{Z}/16\mathbb{Z}$ is uniquely maximal

but

$6\mathbb{Z}/12\mathbb{Z} \not \subseteq 4\mathbb{Z}/12\mathbb{Z} \not \subseteq 3\mathbb{Z}/12\mathbb{Z}$ $<br /> \not \subseteq 2\mathbb{Z}/12\mathbb{Z}$
ie there is not a maximal ideal that contains all those ideals

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