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I’m trying to show this by showing that $\mathbb Z[x]/I$ is a field. Since in this ring $x=-1$ and so $x^2=1$, but also $x^2=-1$ from which it follows that $1=-1$ and so it is probably isomorphic to $\mathbb F_2$? But I can’t find a homomorphism from $\mathbb Z[x]\ \longrightarrow\ \mathbb F_2$ with kernel $I$.

Servaes
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Loobear23
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3 Answers3

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Very useful exercise: If $R$ is a commutative ring and $r_1,\ldots,r_n\in R$, then $$R/\langle r_1,\ldots,r_n\rangle\cong (R/\langle r_1\rangle)/\langle\bar{r_2},\ldots,\bar{r_n}\rangle.$$

Applying this here makes the exercise very easy; we have $R=\Bbb{Z}[x]$ and $r_1=x+1$ and $r_2=x^2+1$. Then $$\Bbb{Z}[x]/\langle x+1,x^2+1\rangle\cong(\Bbb{Z}[x]/\langle x+1\rangle)/\langle\overline{x^2+1}\rangle.$$ Of course $\Bbb{Z}[x]/\langle x+1\rangle\cong\Bbb{Z}$ by mapping $x$ to $-1$. Then $x^2+1$ is mapped to $(-1)^2+1=2$ and so $$(\Bbb{Z}[x]/\langle x+1\rangle)/\langle\overline{x^2+1}\rangle\cong\Bbb{Z}/\langle2\rangle=\Bbb{F}_2.$$ This is a field, and so this shows that the original ideal is maximal.

Alternatively, you mention that you already suspect that the quotient is isomorphic to $\Bbb{F}_2$, but cannot find a homomorphism $\Bbb{Z}[x]\ \longrightarrow\ \Bbb{F}_2$ with kernel $I$. Note that such a homomorphism is determined entirely by where $x$ is mapped. So $x$ must map to some element of $\Bbb{F}_2$ such that $x+1$ and $x^2+1$ are mapped to $0$. There aren't many candidates; you just have to check that this does indeed work.

Servaes
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Note that $I$ is maximal iff $\mathbb Z[x]/I$ is a field.

But $\mathbb Z[x]/(x+1,x^2+1)\cong \mathbb Z[-1]/((-1)^2+1)\cong\mathbb Z/2\mathbb Z=\mathbb F_2$ which is a field.

user26857
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Bach
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    Why does the first isomorphism hold? What does $\mathbb Z[-1]/2$ represent? – Loobear23 Nov 04 '20 at 22:56
  • @Loobear23 Because $x$ is evaluated to be $-1$. $\mathbb Z[-1]$ represents the integer ring $\mathbb Z$ adjoined by $-1\in\mathbb Z$ which is still $\mathbb Z$. $(2)$ represents the principal ideal generated by $2$ in $\mathbb Z$ which gives you $2\mathbb Z$. – Bach Nov 04 '20 at 22:58
  • I think I’m maybe supposed to be using the first isomorphism theorem. Are there any simple homomorphisms? – Loobear23 Nov 04 '20 at 23:01
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    @Loobear23 I don't think using the first isomorphism theorem(well, at least directly) would make this problem any easier. But I am indeed using the third isomorphism theorem indirectly here. If you want a step-by-step explanation, you can refer to Servaes's answer. – Bach Nov 04 '20 at 23:29
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I would put it this way: \begin{align} \mathbf Z[X]\bigm/(X+1,X^2+1)&=\mathbf Z[X]\bigm/(X+1,X^2+1\bmod X +1))\\ &=\mathbf Z[X]\bigm/(X+1,2)\simeq\mathbf Z/2\mathbf Z[X]/(X+1)\\ &\simeq\mathbf Z/2\mathbf Z. \end{align}

Bernard
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