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Task is real simple to understand: I need an example of two fields, E and F, such that their additive parts are isomorphic, and their multiplicative parts are also isomorphic, but fields themselves are not isomorphic.

What I was thinking are two things: First, if E and F would be finite, they would be vector fields over the same field, of the same dimension, and thus linear algebra tells us this is isomorphic. So, counterexample has to be among infinite fields.

Now, I got stuck there. Counterexample cannot be C and R, so we have to search among $Q(\alpha_1,...., \alpha_n)$, but then it seems that we have vector fields of dimension n over field Q, which are again isomorphic.

Now, you see, I must be wrong, because professor who asked this is a real pro in algebra, and he would not allow any mistake. This is just curiosity, because he asked just to think about this, this is no homework for mark or anything. Thanks for any help!

nikola
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    "so we have to search among $\mathbb{Q}(\alpha_1,\dots,\alpha_n)$" -> there are fields that are neither $\mathbb{C}$ or $\mathbb{R}$, nor $\mathbb{Q}(\alpha_1,\dots,\alpha_n)$. Also, $\mathbb{Q}(\alpha_1,\dots,\alpha_n)$ doesn't have dimension $n$ over $\mathbb{Q}$, and even if that was the case, that would go in your direction since you want the additive groups to be isomorphic. – Captain Lama Apr 24 '16 at 07:32

2 Answers2

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Take $K=\mathbb{Q}(X)$ and $L=\mathbb{Q}(X,Y)$.

They are both $\mathbb{Q}$-vector spaces of countable dimension, so they are isomorphic as $\mathbb{Q}$-vector spaces, hence as additive groups.

Now we use the fact that $\mathbb{Q}[X]$ and $\mathbb{Q}[X,Y]$ are UFD with sets of irreducible of the same cardinality, and isomorphic unit groups.

Indeed, let $S$ (resp. $T$) be a set of representative of irreducibles of $\mathbb{Q}[X]$ (resp. $\mathbb{Q}[X,Y]$). Then $\Phi: \mathbb{Q}^* \oplus\left(\bigoplus_S \mathbb{Z}\right)\to K^*$ defined by $\Phi\left( u, (n_P)_{P\in S}\right) = u\prod_{P\in S}P^{n_P}$ is an isomorphism. Likewise, $L^*$ is isomorphic to $\mathbb{Q}^* \oplus\left(\bigoplus_T \mathbb{Z}\right)$, and thus since $S$ and $T$ are equipotent, $K^*$ and $L^*$ are isomorphic.

But $K$ and $L$ are not isomorphic fields. Indeed, $K$ has transcendance degree $1$ over its prime field, while $L$ has trancendance degree $2$.

I must add that while it's a good idea to take seriously what your professor says, everyone makes mistakes. Even the greatest mathematicians of all time made mistakes.

Captain Lama
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I'm afraid I have insufficient reputation to comment, but an example is provided in the question 'Examples of non-isomorphic fields with isomorphic group of units and additive group structure'.

The example given is $\mathbb Q(i\sqrt 2)$ and $\mathbb Q(i \sqrt 7)$.

Josh Hunt
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