Re: New paper, algebraic integers, Galois Theory

From: James Harris (jstevh_at_msn.com)
Date: 10/06/04 

Date: 6 Oct 2004 15:06:19 -0700

Second correction to Section 3:

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I. First section

Start with

P(m) = f^2 ((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2 + u^3 f)

with the factorization

P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)

and note that at

m=0, P(0) = u^2 f^2(3x + uf),

which gives you terms that do not vary as m varies.

So what about (a_1 x + uf), (a_2 x + uf), and (a_3 x + uf)?

(a_1 x + uf)(a_2 x + uf)(a_3 x + uf) = u^2 f^2 (3x + uf)

which shows that at least two of the a's have to equal 0 at m=0, while
one equals 3.

Since, at m=0, two of the a's have to equal 0, it's convenient to just
arbitrarily select a_1 and a_2 as those two.

Then you have uf for the first, uf for the second and 3x + uf for the
third as terms that do not vary when m varies.

Now then, if m=1, what are the *constant* terms?

They are uf, for the first, uf for the second, and 3x + uf for the
third.

That's logical because they do not vary with m, so if m=1003909273,
what are the constant terms?

They are uf, for the first, uf for the second, and 3x + uf for the
third.

Now divide f^2 from both sides, which gives

P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2 + u^3 f

P(m)/f^2 = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)/f^2

and you note that P(0)/f^2 = u^2(3x + uf), which means that now your
constant terms are u for the first, u for the second and 3x + uf for
the third.

Now then, if m=1, what are the constant terms now?

They are u for the first, u for the second, and 3x + uf for the third.

If m = 2938479378, what are the constant terms now?

They are u for the first, u for the second, and 3x + uf for the third.

How can the constant terms of the first two go from uf to u?

They must be divided by f.

Now, the constant term of a_1 x + uf, is uf, but when f^2 is divided
from P(m), it is u; therefore, a_1 x + uf is divided by f, and you
have

a_1 x/f + u

and the constant term of a_2 x + uf is uf, but when f^2 is divided
from P(m), it is u; therefore, a_2 x + uf is divided by f, and you
have

a_2 x/f + u

while the constant term of a_3 x + uf is 3x + uf, and after f^2 is
divided off, it is 3x + uf, so you have

a_3 x + uf

so, dividing P(m) by f^2 gives

P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf).

There is no way to mathematically argue with the result.

II. Second section

Now take

P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf)

and multiply inside the parentheses by f^2/(a_1 a_2 a_3), and outside
by f^2(a_1 a_2 a_3) and you have

P(m)/f^2 = ((a_1 a_2 a_3)/f^2)(x + uf/a_1)(x + uf/a_2)(x + uf/a_3)

and since a_1 a_2 a_3 = f^2(m^3 f^4 - 3m^2 f^2 + 3m), that is

P(m)/f^2 =

        (m^3 f^4 - 3m^2 f^2 + 3m)(x + uf/a_1)(x + uf/a_2)(x + uf/a_3).

Now consider the case that m, f, and u are algebraic integers, then I
have the ratios of algebraic integers:

uf/a_1, uf/a_2, and uf/a_3,

and now let

v_1/w_1 = uf/a_1, v_2/w_2 = uf/a_2, and v_3/w_3 = uf/a_2

where the v's and w's are algebraic integers in each case coprime to
each other.

Making the substitutions I have

P(m)/f^2 =

     (m^3 f^4 - 3m^2 f^2 + 3m)(x + v_1/w_1)(x + v_2/w_2)(x + v_3/w_3).

And I have from before that

P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1 + mf^2) xu^2 + u^3 f

so

(m^3 f^4 - 3m^2 f^2 + 3m)(v_1 v_2 v_3)/(w_1 w_2 w_3) = f

as that is the last coefficient from the last term u^3 f, which proves
that

(m^3 f^4 - 3m^2 f^2 + 3m) has w_1, w_2 and w_3 as factors, so let

(m^3 f^4 - 3m^2 f^2 + 3m) = w_1 w_2 w_3

then I have

P(m)/f^2 = (w_1 x + v_1)(w_2 x + v_2)(w_3 x + v_3)

but I still have that

P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf).

III. Third section

So, even if a_1/f is not an algebraic integer, you can find w_1 an
algebraic integer.

But if a_1/f is an algebraic integer and w_1 is not, they cannot be
equal.

But I have

P(m)/f^2 = (w_1 x + v_1)(w_2 x + v_2)(w_3 x + v_3)

and

P(m)/f^2 = (a_1 x/f + u)(a_2 x/f + u)(a_3 x + uf)

so how do you reconcile a case where a_1 x/f is not an algebraic
integer?

There must exist z_1, z_2, and z_3 such that

w_1 = (a_1 x z_1)/f, w_2 = (a_2 x z_2)/f and w_3 = a_3 x z_3

and z_1 z_2 z_3 = 1,

so algebraically the z's are units, but z_1, z_2 and z_3 are not units
in the ring of algebraic integers, since a_1/f is not.

Challenging Section 1 is challenging algebra itself.

That leaves Galois Theory to be handled next.

James Harris

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