Examples of non-isomorphic fields with isomorphic group of units and additive group structure

YACP mentions in a comment that:

There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong (L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$

Can someone provide an example?

I have found the following thread. But there, the underlying structure is a ring. Hence, an answer to the current question is automatically an answer to the old question as well. EDIT: Sorry, the other question asks for finite commutative ring, and this won't be possible in the case of fields…

• $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$? – user27126 Jan 1 '14 at 21:06
• @Sanchez This can be a good example as one knows that $\mathbb Z[\sqrt 2]$ and $\mathbb Z[\sqrt 3]$ are the rings of algebraic integers of $\mathbb Q(\sqrt 2)$ and $\mathbb Q(\sqrt 3)$, respectively and both are euclidean. But their groups of units are not so easy to handle. (Anyway, they are both isomorphic to $\{\pm 1\}\times\mathbb Z$ if I'm not wrong.) That's why I've preferred imaginary quadratic number fields. – user26857 Jan 1 '14 at 22:00

An example can be the following: $\mathbb Q(i\sqrt 2)$ and $\mathbb Q(i\sqrt 7)$.
With respect to the multiplicative groups, note that both are isomorphic to $\{\pm1\}\times\mathbb Z^{(\mathbb N)}$ (the last denotes a countable direct sum of copies of the group of integers).