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-mathématiques.net that very question was asked, and it has been shown the following.

Define a *Siamese* to be a quadruplet , where is a field and is a ring.

Then, if there exists an element of finite order , then , where is a prime number, and . 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 .

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

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Tags: algebra, field theory, group theory, ring theory

This entry was posted on December 4, 2007 at 10:53 pm and is filed under math.RA, mathematics. You can follow any responses to this entry through the RSS 2.0 feed.
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