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In class we said that, as a consequence of Birkhoff's theorem, the theory of fields is not axiomatizable only by equations. In particular, we saw that the product of two fields in general is just a ring. However I don't understand why if we use a language with $(+,\cdot,-,\frac {1}{}, 1,0)$, where $+$ and $\cdot$ are the binary operations of sum and product, $-$ and $\frac 1 {} $ are the unary operations that give the inverses, and $1,0$ are the two usual constants. It seems to me that in this language the theory of fields is axiomatizable adding to the axioms of a ring (that are all equations) the axiom $\frac 1 x \cdot x=1$, that is an equation too. Probably I'm missing some hypothesis on the language in the Birkhoff's theorem, but I don't see any of them. Thaks for any clarify

Dr. Scotti
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1 Answers1

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The problem is that $(x\mapsto 1/x)$ is only a partial operation in fields (division by $0$ is undefined), that cannot be captured in universal algebra where all operations have to be total.

Jonas Frey
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  • No, this is not the problem. The problem is that the axiom $x \cdot \frac{1}{x} = 1$ can't be satisfied if $x = 0$ no matter what value we assign $\frac{1}{0}$! I went into more detail here: https://math.stackexchange.com/questions/3756136/why-is-the-category-of-fields-seemingly-so-poorly-behaved/4941415#4941415 – Qiaochu Yuan Jul 04 '24 at 01:06