Showing two field extensions are equal

Question

Let a,b $\in$ $\mathbb Q$ with b nonzero. Show that $\mathbb Q$($\sqrt a$)=$\mathbb Q$($\sqrt b$) if and only if $\exists$ c $\in$ $\mathbb Q$ such that a=b$c^2$.
I am confused on how it is possible to show that $\mathbb Q$($\sqrt a$)=$\mathbb Q$($\sqrt b$) by assuming such a c exists. If a=b$c^2$ then $\mathbb Q$($\sqrt a$)=$\mathbb Q$($\sqrt (bc^2)$)=$\mathbb Q$(|c|$\sqrt b$)=$\mathbb Q$($\sqrt b$) since c $\in$ $\mathbb Q$, but I think this falls apart if c=0. Is this approach flawed or must c be nonzero for this to hold?

Answer 1

As Roger points out in the comments, we need an additional condition. It is enough to require that $c$ be non-zero, or alternately that $a$ be nonzero. I suspect that the latter was intended.