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

Question

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…

Answer 1

An example can be the following: $\mathbb Q(i\sqrt 2)$ and $\mathbb Q(i\sqrt 7)$.

It's well known that these fields are not isomorphic: quadratic fields are isomorphic if and only if they are equal.

It is obvious that their additive groups are isomorphic.

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).