I am reading from Wikipedia's article on Integral Element [*here*](https://en.wikipedia.org/wiki/Integral_element), where it says that,

> 
> ... an element $b$ of a commutative ring $B$ is said to be integral over $A$, a subring of $B$, if there are $n \geq 1$ and $a\_j \in A$ such that
>  $$b^n + a\_{n-1}b^{n-1} + ... + a\_1b + a\_0 = 0.$$
>  This is to say, $b$ is a root of a monic polynomial over $A.$
> 
> 
> 

It goes on stating that the Gaussian Integers, a complex numbers of the form
$a + b\sqrt {-1},$ with $a, b \in \mathbb Z,$ are integral over $\mathbb Z.$ Here are my two questions:

(a) I have been tinkering with this example for more than one hour but could not come up with a monic polynomial having $a + b\sqrt {-1}$ as its root. Could anybody give me a hand? Do I convert it to Euler Formula $e^{ix}$ first?

(b) Do you have any other ***simple*** example of this integrality over $\mathbb Z$ befitting a beginner's knowledge?

Thank you for your time and help.