Partial isomorphisms between fields

Question

An isomorphism of fields $F$ and $E$ is defined by a function $\phi$ which is an isomorphism of groups $\phi:(F,+)\rightarrow (E,+)$ that restricts to an isomorphism of groups $\phi:(F\setminus\lbrace 0 \rbrace,\cdot)\rightarrow (E\setminus\lbrace 0 \rbrace,\cdot)$. In my current work, I have a bijection $\phi$ between fields, say $E$ and $F$ such that $\phi$ restricts to a multiplicative isomorphism, but it remains unclear whether $\phi$ is indeed a field isomorphism.

My question is essentially, given a map between two fields that restricts appropriately to a multiplicative isomorphism, are there examples where the map is not a field isomorphism? My initial thought was that there should be some examples, but I am struggling to find any.

Answer 1

Well, my idea is to consider the inversion, with a correction, so take a field $E$ and the map $\phi:E\rightarrow E$ defined as $$\phi(x)=\left\{\begin{array}{ll} x^{-1} & x\neq0\\ 0 & x=0\end{array}\right.$$this is a 1-1 correspondence $E\rightarrow E$ and is a multiplicative isomorphism restricted to $E\setminus\{0\}$ (here is essential $E\setminus\{0\}$ being abelian as a multiplicative group), but is clearly not a field isomorphism

Answer 2

Consider the domains $\mathbb{C}[x]$ and $\mathbb{C}[x,y]$. They have isomorphic unit groups (nonzero complex numbers), and are both UFDs with uncountably many irreducibles; so the multiplicative structure of both of them is isomorphic. That is, if we let $f$ be an arbitrary bijection between the set of monic irreducibles in $\mathbb{C}[x]$ and the set of monic irreducibles of $\mathbb{C}[x,y]$, we can extend $f$ to a multiplicative isomorphism between $\mathbb{C}[x]$ and $\mathbb{C}[x,y]$ by having it be the identity on $\mathbb{C}$ and act as $f$ on the irreducibles. Of course, in general this will not be a ring isomorphism, since $\mathbb{C}[x]$ is a PID but $\mathbb{C}[x,y]$ is not.

Now we can extend this map into a bijection between $\mathbb{C}(x)$ and $\mathbb{C}(x,y)$ in the obvious way: maintain the identity on the units, and just extend it to include negative exponents of the monic irreducibles.

This gives us a bijection between $\mathbb{C}(x)$ and $\mathbb{C}(x,y)$ that is multiplicative, but that is not a field isomorphism. In fact, the two fields are not isomorphic, since they have different transcendental degree over $\mathbb{C}$.

(This might be a slight improvement over Alessandro’s answer, since in that case it just so happens the two fields are isomorphic, just not as an extension of the map given; here, there simply is no field isomorphism at all.)

P.S. this is somewhat related to a question I had asked in math.overflow some time ago, related to whether you could have (finite) rings $R$ and $S$ that were isomorphic as abelian groups and as multiplicative semigroups, but not as rings; I had used the example of $\mathbb{C}[x]$ and $\mathbb{C}[x,y]$ in the infinite case. That question is here.