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The notion of a field (a commutative ring $R$ with $0\neq 1$ and $R^\times=R-\{0\}$) seems to fit uncomfortably into modern algebra. To see what I mean, consider the following statements:

  1. The concept of a field seems necessary for the foundation of geometry. For example, the "Steinitz Exchange Lemma" (proving that dimension of a vector space is well-defined) seems to require a field. Dimension of a vector space, in turn, seems to underlie all other notions of dimension in mathematics.

  2. The category of fields is not much of a category. It doesn't have products or coproducts. Every morphism is a monomorphism. It's not connected (there is one connected component for each characteristic). Free fields don't exist. In general, fields do not form an "equational theory" so they don't behave like an "algebraic category".

So what is a field, really? Here are a couple suggestions to get you started:

  • A single field $K$ is something like the category of sets. This is suggested by the analogy between the bifunctor $\mathrm{Hom}_{\mathcal{C}}(-,-):\mathcal{C}^{\mathrm{op}}\times\mathcal{C}\to\mathsf{Set}$ on a category $\mathcal{C}$ and a bilinear form $\langle -,-\rangle:V\times V\to K$ on a $K$-vector space $V$. It also agrees with the use of the "ground field" in algebraic geometry, which is something like the "universe of discourse".

  • The construction of the splitting field doesn't have an UMP because fields have too many automorphisms. So it seems that "Galois theory" is what makes fields special. Perhaps a formalization of Galois theory can help explain what fields really are.

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  • $\begingroup$ If I'm not mistaken, Grothendieck generalized Galois Theory in quite a broad way (not in the Big Apple though), so I'm not sure that a formalization of Galois theory is needed. Still, I think Marc Krasner (spelling?) did study an abstract Galois theory, I kind of remember having read a PDF entitled "Théorie de Galois abstraite" that could be of interest if you read French. $\endgroup$
    – Sylvain JULIEN
    Commented May 20, 2016 at 19:59
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    $\begingroup$ The theory of fields fits into the context of coherent geometric logic in topos theory. $\endgroup$
    – Benjamin Steinberg
    Commented May 20, 2016 at 21:41
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    $\begingroup$ @BenjaminSteinberg That's true for the notion of "discrete field" as discussed in the nLab, ncatlab.org/nlab/show/field, but as it turns out that's a rather restrictive notion. (E.g., the Dedekind reals in a topos do not form a field in this sense.) A somewhat vexing issue is that constructively speaking, there is a variety of notions. $\endgroup$
    – Todd Trimble
    Commented May 22, 2016 at 17:01
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    $\begingroup$ This question is relevant: mathoverflow.net/questions/3003 $\endgroup$
    – Neil Strickland
    Commented May 25, 2016 at 9:51

2 Answers 2

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Fields are the simple (no nontrivial quotients) commutative rings. Grothendieck told us to work in nice categories with nasty objects rather than nasty categories with nice objects; fields are the nice objects, and the nice category they live in is all commutative rings. You want to work in all commutative rings and sometimes study fields the same way you want to work in all modules and sometimes study simple modules.

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    $\begingroup$ I'm not sure your characterisation of local rings is correct. What if I invert a single element? $\endgroup$
    – Zhen Lin
    Commented May 20, 2016 at 22:07
  • $\begingroup$ @Zhen: hmm, yeah, I was being silly (e.g. $F[[x, y]]$ can localize to $F[[x, y]][y^{-1}]$ which is another local ring). I think there ought to be a true statement along these lines, which might require using all epimorphisms rather than just localizations. $\endgroup$
    – Qiaochu Yuan
    Commented May 20, 2016 at 22:22
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Related to the previous answer but different - in abelian categories, (not necessarily commutative) fields are endomorphism rings of simple objects (just as local rings are endomorphism rings of indecomposable objects). From this point of view it is even somehow not natural to consider their homomorphisms: I cannot think of any kind of natural data between two objects which would induce a homomorphism between their endomorphism rings. The natural kind of morphisms to consider in this context are bimodules - hom between objects is a bimodule over their endomorphism rings.

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