Galois extensions for cyclic group:Z3

This article discusses Galois extensions whose Galois group is isomorphic to the cyclic group of order three.

Example of a cyclotomic extension
Consider the cubic extension over $$\mathbb{Q}$$ obtained by adjoining $$2\cos(2\pi/7)$$. The minimal polynomial for this is:

$$x^3 + x^2 - 2x - 1$$.

This extension is a Galois extension. To see this, it suffices to show that adjoining one of the roots of this irreducible cubic is enough to adjoin all the roots. Indeed, the other two roots of this polynomial are $$2\cos(4\pi/7)$$ and $$2\cos(6\pi/7)$$, and both of them are expressible as polynomials in $$2\cos(2\pi/7)$$.