How to prove if a set is a field or not?

Question

For this question, I'm not sure how to prove whether this set is a field or not. I know that if the set is a field it needs to fill in all the requirements, like the commutative, associative, and distribution laws, as well as the additive and multiplicative principle, which it looks like it does. I'm not sure how to further prove this. Can anyone please help me out?

Is the set $\mathbb{Q}[\sqrt{7}] := \{a + b \sqrt{7}\mid a, b ∈ Q\}$ with addition and multiplication defined by $(a + b \sqrt{7}) ⊕ (c + d \sqrt{7}) =(a + c) + (b + d) \sqrt{7}$ for all $a, b, c, d ∈ \mathbb{Q}$, $(a + b \sqrt{7} ⊗ (c + d \sqrt{7}) =(ac + 7bd) + (ad + bc) \sqrt{7}$ for all $a, b, c, d ∈ \mathbb{Q}$, a field?

Answer 1

These are the ordinary addition and multiplication in $\mathbf C$, and $\mathbf Q\bigl[\sqrt 7\bigr]$ is a subset thereof. So all you have to prove is

1. $\;\mathbf Q\bigl[\sqrt 7\bigr]$ is a subring of $\mathbf C$, i.e. it is stable by addition and multiplication, and $1\in\mathbf Q\bigl[\sqrt 7\bigr]$.
2. The inverse of a non-zero element $a+b\sqrt 7$ $\;(a,b\in\mathbf Q)$ is an element of $\mathbf Q\bigl[\sqrt 7\bigr]$.