Determine the field properties that are satisfied by B, Is B a field?

Question

Let B be the set of all irrational numbers together with the numbers 0, 1, and -1. Let addition and multiplication be defined on B in the same way they are defined for real numbers. Determine the field properties that are satisfied by B. Is B a field?

Answer 1

Since $\sqrt{2}$ is irrational, $\sqrt{2} \in B$.

However, $\sqrt{2} \cdot \sqrt{2} = 2 \notin B$. Therefore $B$ is not closed under multiplication and hence $B$ is not a field.

Answer 2

As an alternative you have that $2-\sqrt{2}, \sqrt{2}$ are both irrational and their sum is $2$ which is not in $B$ so it is not closed under addition either.