Friday, July 10, 2009


JSH: Question about Galois Theory

I have some questions about Galois Theory.

In the ring of algebraic integers consider the simple quadratic:

x^2 - 6x + 35 = 0

The solutions are non-rational, and traditional view by Galois Theory is that the factors of 35, for instance, 7, split in ways I think that have to do with the class number, but let y = x - 35, then x = y + 35, so:

(y+35)^2 - 6(y+35) + 35 = 0

which is:

y^2 + 64y + 1050 = 0.

Now exactly one solution for x should be a factor of exactly one solution for y, so do the class numbers match? (Am I using the phase "class number" correctly? Sorry if not.)

Can you show solutions where y/x is an algebraic integer?

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