# Are natural numbers a field with alternate addition and multiplication

I've just started P.Halmos' book "Finite Vector Spaces", and at the first chapter, after defining the axioms of a field, there is an exercide that asks: if we consider the non negative integers, can we redefine addition and multiplication so that they form a field? Can't we, since $\mathbb{N}$ is countable just like $\mathbb{Q}$ that is a field, use some sort of coding to represent the field $\mathbb{Q}$ with it's operations with the natural numbers?

• Yes, that would work. Any infinite set, or any finite set of prime power size $>1$, can be given the structure of a field. – Noah Schweber Oct 26 '16 at 23:56
• I was always under the impression that they were. – Ninosław Brzostowiecki Oct 26 '16 at 23:59
• Can someone give an example of how that would look like? – Fede Poncio Oct 27 '16 at 0:00

• Suppose you want to put a group structure on $\mathbb{N}$. Since $\mathbb{N}$ has the same cardinality as $\mathbb{Z}$, there's some bijection $\phi$ between them. Then you can interpret $\mathbb{N}$ as a group where $a + b$ is interpreted as $\phi^{-1}(\phi(a) + \phi(b))$. – Nitin Oct 27 '16 at 0:22
• Your question was "if we consider the non negative integers, can we redefine addition and multiplication so that they form a field?" The answer to your question is yes as you just explained. The field constructed out of the nonnegative integers will be isomorphic to $\mathbb{Q}$. – Not a grad student Oct 27 '16 at 0:51