1. ## Abstract Algebra question

Let R = {all real numbers}. Then <R,+> is a group. (+ is regular addition)

Let H = {a|a is an element of R and a^2 is rational}.

Is H closed with respect to the operation?
Is H closed with respect to the inverse?
Is H a subgroup of G?

I know what it means to be closed with respect to the operation and inverse but I'm not sure how to show it. Thanks.

2. Originally Posted by Iceflash234
Let R = {all real numbers}. Then <R,+> is a group. (+ is regular addition)

Let H = {a|a is an element of R and a^2 is rational}.

Is H closed with respect to the operation?
Is H closed with respect to the inverse?
Is H a subgroup of G?

I know what it means to be closed with respect to the operation and inverse but I'm not sure how to show it. Thanks.
Well, essentially you need to prove that if $a^2 \in \mathbb{Q}$ and $b^2 \in \mathbb{Q}$ then $(a+b)^2 \in \mathbb{Q}$. So simply expand $(a+b)^2$ and see where this gets you. If you don't quite see what is going on, try a couple of examples. For instance, $\sqrt{2}$ and $\sqrt{3}$.

You're inverses here are the negatives: $-a$ is the inverse of $a$. So if $a^2 \in \mathbb{Q}$ then...?