Sunday, December 06, 2009


JSH: Losing Galois Theory

One of the more depressing things that happened several years ago was that a line of amateur mathematical research took away the usefulness of Galois Theory. Turns out it's now trivial to prove, with a simple demonstration:

Try to divide off the 7:

7(175x^2 - 15x + 2) = (5a_1(x) + 7)(5a_2(x) + 7)

where the a's are roots of

a^2 - (7x-1)a + (49x^2 - 14x) = 0 in the ring of algebraic integers.

You may have seen me post that before as a challenge to mathematicians and string theorists (who happen to often be heavily into mathematics), but you may not understand its full significance—or how it takes away Galois Theory, so here is the rest.

Try to divide off the 7 in the most general way possible, consider functions,

w_1(x) and w_2(x), such that:

w_1(x)*w_2(x) = 7, and dividing off the 7 above gives:

175x^2 - 15x + 2 = (5a_1(x)/w_1(x) + 7/w_1(x))(5a_2(x)/w_2(x) + 7/w_2(x))

and suppose that they are factors such that

a_1(x) = b_1(x)*w_1(x), and a_2(x) = b_2(x)*w_2(x), then:

175x^2 - 15x + 2 = (5b_1(x) + w_2(x))(5b_2(x) + w_1(x))

and you may think that all is ok, but notice—a residue of 7 remains in the form of the factors w_1(x) and w_2(x).

The freaky bastards don't want to go away!!!

They are ghosts that remain in the thing.

But for those who know their Galois Theory that is unacceptable, as the class number uniquely holds those factors so it really is saying that SOMETHING is left, even though the 7 is gone from the left hand side—and that something can't just disappear by the rules of Galois Theory—in the ring of algebraic integers.

It is there permanently. But that is nonsensical. The 7 has divided off, so what can be left?

With an example from integers:

7(x^2 + 3x + 2) = (7x + 7)(x+2),

divide off the 7: x^2 + 3x + 2 = (x+1)(x+2). Done.

And if someone told you that a trace of the 7 had to remain you'd call them an idiot.

You may know that physicists seem to think that Galois Theory has usefulness in physics with group theory, but if you can comprehend the mathematics above, you now know it can't, because Galois Theory is about unit factors, which for that reason can always just divide off and disappear.

Unless you believe in math ghosts…

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