JSH: Losing Galois Theory

Subject:	JSH: Losing Galois Theory
Date:	Sun, 6 Dec 2009 09:42:28 -0800 (PST)

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....

James Harris