Two questions on the Gaussian integers

Question

I have two questions on the Gaussian integers.

Is any element in $\mathbb{Z}[i]$ the root of a monic polynomial with coefficients in $\mathbb{Z}$?
Conversely, does any element in $\mathbb{Q}(i)$ that is the root of a monic polynomial with coefficients in $\mathbb{Z}$ lie in $\mathbb{Z}[i]$?

i.e. $,w,$ is a root $,(x-w)(x-w') = x^2 - (w!+!w') x + ww',$ where $,w'$ is the conjugate of $w,,$ so $,w!+!w',, ww'\in \Bbb Q.,$ For the 2nd question use Gauss's Lemma — Bill Dubuque, Jul 26 '16 at 16:42

score 0 · Answer 1 · answered Jul 26 '16 at 16:59

1) $a+bi$ is a root of $x^2-2ax+(a^2+b^2)=0$.

2) For the converse, consider the quadratic $x^2+bx+c$, where $b$ and $c$ are integers. If the roots are rational, they are divisors of $c$. Now we deal with the case the roots are non-real. From the quadratic formula, if the roots are non-real and in $\mathbb{Q}(i)$, they are of the shape $\frac{s\pm ti}{2}$, where $s$ and $t$ are integers.

Then $\frac{s^2+t^2}{4}=c$. Thus $s^2+t^2\equiv 0\pmod{4}$. This forces $s$ and $t$ to be even, and we are finished.

Two questions on the Gaussian integers

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