Friday, October 06, 2006

 

Checking splitting fields, extreme mathematics at work

I've been brainstorming the last few days with a fascinatingly simple factorization tool, where I've tossed out various ideas without checking carefully because that is what brainstorming is.

That work has paid off as I've gone from a vague sense that this idea could say something about class number to focusing now on splitting fields and realizing it offers a way to check them.

I'll be using some more complicated expressions than before, while the basic equations are the same:

With

x^2 - ax + n = 0

and

y^2 - by + n = 0

I have

(y + x - a)*(y-x) = (b-a)*y

as my generalized factorization result.

So I want to compare splitting fields and for that purpose I will use

y_1 = (5 + sqrt(17))/2*(7 - sqrt(41))/2

y_2 = (5 - sqrt(17))/2*(7 + sqrt(41))/2

where I'm working backwards instead of starting with quadratics. Those solutions give me n=4, but now I need to figure out b:

b = y_1 + y_2

and now I'm free to pick a=4, as I think I can let x be an integer and still get the result I want, as then x=2.

Then

(y - 2)*(y - 2) = (b-4)*y

and my guess—why I chose y_1 and y_2 as I did in terms of sign—is that I managed to pick in such a way that both y_1 and y_2 have 2 as a factor!!!

Then 2 will be a factor when you multiply the roots by two's of the polynomial with integer coefficients that has (y + x - a) and (y-x) as roots!!!

The trick here is forcing y to always have 2 as a factor by careful use of signs.

If my hunch is correct then the result using those values shows the uselessness of splitting fields when it comes to telling you what the factors actually are, as in how 2 factors between

(5 + sqrt(17))/2 and (7 + sqrt(41))/2

where I had a feeling this factorization result of mine could be used for this, but just had to figure out how to do it.

That's how extreme mathematics works—throw something out there and hope a good idea will eventually come to you.

And this one—if I'm right—shows directly how Galois Theory as currently taught is wrong.





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