## Archive for December, 2007

December 5, 2007

That’s what folks at Herculanum have found. It’s great to see excavations are ongoing these days, let’s hope they’ll have a look at the papyri too in the near future.

I’m also eargerly anticipating the planned robot-based exploration of the secret shafts in the great pyramid, but it’s been two years since news appeared on the issue. I hope this is not due to a lack of funds…

Anyway, so many things, so little time.

### Groups, rings: what’s next?

December 4, 2007

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 $(K,+,\times, \diamondsuit)$, 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.