Proving the set is a field

[summerset353]

Let A be the set of all numbers of the form a+ b[sqt(2)], where a and b are arbitrary rational numbers. Let addition and multiplication be defined on A in the same way they are defined for real numbers. Prove that the set A is a field.

 

[daon]

If every element is of the form $\displaystyle a+b\sqrt{2}$ with $\displaystyle a,b \in \mathbb{Q}$, you need to show:

1) closure under addition, multiplication
2) associativity and commutativity for both as well
3) additive inverse for all elements
4) for a and b not both zero, each has a multiplicative inverse
5) distributivity

What have you tried? I will not do it for you, but help you.