Proof that $M \;$is a field

Question

Let $\; p,d \,$ be prime numbers I wanna proof that $\; M:=\{a_o+a_1\sqrt p +a_2\sqrt d +a_3\sqrt {pd}:a_o,a_1,a_2,a_3 \in \mathbb Q \}\;$ is a field. The only thing I am having trouble with is finding the inverse of $p \in M$. Is it true that $\;M\;$ is a field and if so how would one find the inverse of an element. Give me a little hint.