proving that $a + b \sqrt {2} + c \sqrt{3} + d \sqrt{6} $ is a subfield of $\mathbb{R}$ The question is given below:



My questions are:
1- How can I find the general form of the multiplicative inverse of each element?
2-How can I find the multiplicative identity?
3-Is the only difference between the field and the subfield definition is that in the case of a subfield every nonzero element has an additive identity but in the field every element not only nonzero ones?
Could anyone help me in understanding these questions, please?
EDIT:
I have found this solution on the internet:



My question: is that a fully acceptable answer to the question? I guess yes.

Check the field axioms for such expressions. You really need only that the sum and product has the same form, identify the additive and multiplicative inverses; the others follow as you are operating on real numbers. They hold, and the set is clearly a (proper) subset of $\mathbb{R}$. Done.
