TITLE: Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.
QUESTION [15 upvotes]: I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one please check this proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field. Thank you.
Proof: Notice that $$\langle 1 - i \rangle\\ 
\Rightarrow 1 = i\\
 \Rightarrow 2 = 0.$$
Thus all elements of the form $a+ bi + \langle 1 - i \rangle$ can be rewritten as $a+ b + \langle 1 - i \rangle$. But since $2=0$ this implies that the elements that are left can be written as $1 + \langle 1 - i \rangle$ or $0 + \langle 1 - i \rangle$. Thus 
$$
\mathbb Z[i]/ \langle 1 - i \rangle = \{ 0+ \langle 1 - i \rangle , 1 + \langle 1 - i \rangle\}.
$$
This is obviously a commutative ring with unity and no zero-divisors, thus it is a finite integral domain, and hence is a field. $\square$

REPLY [7 votes]: One must also prove that the quotient ring is $\ne \{0\}.\:$ Below is a complete proof.
$\rm\quad  \Bbb Z\stackrel{h}{\to}\, \Bbb Z[{\it i}\,]/(1\!-\!{\it i}\,)\:$ is $\rm\,\color{#0b0}{\bf onto,\:}$ by $\rm\:mod\,\ 1\!-\!{\it i}\,:\ {\it i}\,\equiv 1\phantom{\dfrac{|}{|}}\!\!\!\Rightarrow\:a\!+\!b\,{\it i}\,\equiv a\!+\!b\in \Bbb Z\ $
$\rm\quad  n\in ker\ h\iff 1\!-\!{\it i}\,\mid n\iff\phantom{\dfrac{|}{|_|}}\!\!\!\!\!\!\! \dfrac{n}{1\!-\!{\it i}}\, =\, \dfrac{n\,(1\!+\!{\it i}\,)}2\,\in\, \Bbb Z[{\it i}\,] \iff \color{#c00}2\mid n\ $
$\rm\quad So \ \ \ \Bbb Z[{\it i}\,]/(1\!-\!{\it i}\,)\, \color{#0b0}{\bf =\ Im\:h}\,\cong\, \Bbb Z/ker\:h \,=\, \Bbb Z/\color{#c00}2\,\Bbb Z\, =\, \Bbb F_2\ $  $\ \ $ QED