January 16, 2006

Mushrooms and Mathematics

Unfortunately the two topics mentioned in the title are disjoint, but now I'll have a think and probably come up with something where they're not disjoint another time, anyway,

The Mushroom

Fancying a late-night snack after a hard days maths Steven looked into his fridge in horror, as all that could be found was a half a mushroom and some butter (Steven needed to go shopping), soon a plan was formulated, and only minutes later executed. A blend of herbs and spices were put atop said mushroom before it was gently fried on a low-to-medium heat for 2-to-3 minutes. Never had a Tesco closed cup mushroom had so much attention, the eyes of the world were upon him, bearing down, but he stood firm, soaking up the flavours of the hot butter beneath. It was not to last though, the mushroom had to be eaten, as was duly done so. His last moments were captured on film, which can now be shared by all by the wonders of the internetweb.


In the course of writing my second year essay entitled "Classical problems in Geometry" I have been discussing abstract algebra, and polynomials etc. One thing that has always bothered me about the Fundamental Theorem of Algebra is that it uses

\mathbb{C}, which has all these transcendentals, which, if your polynomial is over \mathbb{Q} are rather useless, and they take up a lot of space, in fact most of the complex plane. Now I realise perfectly well that this has all been worked out before, but the point is I worked this next stuff out for myself, which after being spoonfed maths for over a year, is quite reassuring. We can define the "Algebraic Numbers" as the set of numbers that are solutions to polynomials over \mathbb{Q}, they have cardinality \aleph_0! which is so cool, I mean, really, who needs analysis anyway? Also we can't actually write all of these numbers down as radicals, but we can, of course, as solutions to polynomials, which is still a hell of a lot better than transcendentals.
Galois, you were the man, 'till you got shot.

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