Groups, rings: what’s next?

Have you ever wondered, maybe during an undergraduate algebra course, why nobody speaks about sets equipped with three inner laws? Groups have one law, rings and fields have two, but why not look at things with three or more laws?

Well, over at les-mathé that very question was asked, and it has been shown the following.

Define a Siamese to be a quadruplet (K,+,\times, \diamondsuit), where (K,+,\times) is a field and (K^\times,\times,\diamondsuit) is a ring.

Then, if there exists an element of finite \times-order r\neq 1, then (K,+,\times) \simeq (\mathbb{F}_{p^r},+,\times), where n=p^r-1 is a prime number, and (K^\times,\times,\diamondsuit) \simeq (\mathbb{Z}/n\mathbb{Z},+,\times). In particular the characteristic of a Siamese cannot be 0.

On the other hand, the only infinite Siameses are of characteristic 2, namely the purely transcendental extensions of \mathbb{Z}/2\mathbb{Z}.

I think these results are quite interesting and should be incorporated into any undergraduate algebra curriculum.

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