# I don't really understand polynomial quotient rings

• Mar 17th 2009, 07:35 PM
electricjaguar
I don't really understand polynomial quotient rings
For example, what is the difference between

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

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

I'm having trouble "getting it" if that makes sense. :confused:
• Mar 17th 2009, 07:52 PM
GaloisTheory1
Quote:

Originally Posted by electricjaguar
For example, what is the difference between

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

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

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

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

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

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

Yes b/c $\displaystyle F_2[x]=\mathbb{Z}/2\mathbb{Z} \cong \mathbb{Z}_2$
• Mar 17th 2009, 10:08 PM
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 $\displaystyle \mathbb{Z}/(16)$ contains a unique maximal ideal, but $\displaystyle \mathbb{Z}/(12)$ does not? I think this is the case by the correspondence theorem

Also, I can see why $\displaystyle \mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}$, but why is $\displaystyle \mathbb{R}[x]/(x^2 + x + 1) \cong\mathbb{C}$?
• Mar 18th 2009, 04:01 AM
electricjaguar
I also have a question about $\displaystyle \mathbb{Z}_2[x]/(x^2 + 1)$. Isn't one of its elements $\displaystyle x^2 + 1 + 1 = x^2 + 2 = x^2$ which has a root in $\displaystyle \mathbb{Z}_2$, namely, x = 0? Doesn't this show that this element is reducible and hence the quotient is not a field?

edit: nevermind
• Mar 18th 2009, 08:47 AM
GaloisTheory1
Quote:

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

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

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

but

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