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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.

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