TITLE: Galois Theory by Rotman, Exercise 60, a field of four elements by using Kronecker's theorem and adjoining a root of $x^4-x$ to $\Bbb Z_2$
QUESTION [9 upvotes]: I've been working through Rotman's Galois Theory and am stumped by exercise 60:

Use Kronecker's theorem to construct a field with four elements by adjoining a suitable root of $x^4 - x$ to $\mathbb{Z}_2$.

I can split $x^4 - x$ in $\mathbb{C}[x]$ (which is not a field but which is contained in the field $\operatorname{Frac}(\mathbb{C[x]})$). 
The roots are $F = \{ 0, 1, \frac{-1 - i \sqrt{3}} {2}, \frac{-1 + i \sqrt{3}} {2} \}$.
So what does it mean to adjoin a suitable root to $\mathbb{Z}_2$? I have been unable to construct a four element field using either complex root.
So I went and tried to use the proof of Thm 33 (Galois) which gives a construction. In this case, F is as above and has the required four elements with $p = 2$, $n = 2$, and $q = 4$, and indeed $F \setminus \{0\}$ behaves as desired for multiplication. 
The problem is addition. How is addition defined? Normal addition is not closed, nor do I see how to define it to make it work. Further, given Kronecker's and Galois's theorems, I would assume that addition must be defined as it would be in the containing field.

REPLY [3 votes]: The field is $\mathbb Z_2(\xi)$, where $\xi^2+\xi+1=0$.  It is a $\mathbb Z_2$ -vector space of dimension $2$, with basis $\{1,\xi\}$. Thus has $4$ elements.
It can also be written as $\frac{\mathbb Z_2[x]}{(x^2+x+1)}$.