Questions on the field extension $K = \mathbb{Q}[x]/\langle x^2 − 5\rangle$

Question

Given the field extension $K = \mathbb{Q}[x]/\langle x^2 − 5\rangle$ of $\mathbb{Q}$, and letting $a = [x] ∈ K$;

1) Show $K ≃ \mathbb{Q}(\sqrt5) $ and $[K : \mathbb{Q}] = 2.$

2)Find the minimal polynomial of $a$ over $\mathbb{Q}$ and represent the elements $a^3, (1 + a)^2, (1 + a)^{−1}$ in the form $c + da$, where $c, d ∈ \mathbb{Q}$.

I am quite lost with this question. I struggle with the basic concept of field extensions and calculating their dimesion, and amn't too sure on the form of members of the set K. There are quite a few gaps in my knowledge of this topic so I'd appreciate some clarification. Thanks

Answer 1

Hints:

• Notice that $\alpha = \sqrt{5}$ is a root of $f(x) = x^2 - 5$. Now take any $g(x) \in \mathbb{Q}[x]$ and divide it by $f (x)= x^2-5$. What is the remainder? Next, take $g(\alpha)$, and notice that it can be written in terms of $a + b\alpha$.

• Use the Eiseinstein Criterion to determine whether $f(x) = x^2-5$ is irreducible over $\mathbb{Q}$, from this conclude that the ideal $\langle f(x)\rangle$ is maximal.

• To see that $K \simeq \mathbb{Q}(\alpha)$, use the Isomophism Theorem.

Feel free to ask in the comments.