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## Thursday, October 21, 2010

### Galois Theory (3)

Equations of degree 5 and higher can't be solved. That's what I thought. I knew that certain equations of degree 5 could be solved however like $x^5 - a = 0$. Abel's impossibility theorem ( aka Abel-Ruffini theorem ) says that the quintic can't be solved in general by radicals. In general. What they in fact proved is that some equations can't be solved by radicals. Galois developed a technique for determining if an equation can be solved by radicals or not. This technique was the start of Group- and Galois Theory. A polynomial has a corresponding group, it's so called Galois Group. If this group is solvable, i.e. is a solvable group then the equation can be solved by radicals.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)