1. First the problematic definition:

Algebraic integers are defined to be roots of monic polynomials with
integer coefficient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where
"monic" refers to the leading coefficient.

My assertion is that the over hundred year old definition excludes
numbers that have to be included to keep from having contradiction
i.e. mathematical inconsistency.

2. The important tool I use is a polynomial:

P(m) = 49((2401 m^3 - 147 m^2 + 3m) 5^3 - 3(-1 + 49 m )5 + 7)

The form of the polynomial allows me to factor P(m) into
non-polynomial factors, and the factorization with those factors is

P(m) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7)

where the a's are roots of the following cubic:

a^3 + 3(-1 + 49m)a^2 - f^2(2401 m^3 - 147 m^2 + 3m).

From the factorization I have three factors

g_1 = (5 a_1 + 7), g_2 =(5 a_2 + 7), g_3=(5 a_1 + 7)

but setting m=0, gives me P(0) = 49(3(5) + 7), which fits with the
cubic as at m=0 it gives


a^3 -3a^2 = 0, so a_1 = a_2=0, a_3 = 3,

to show that at m=0, the three factors are

g_1 = 7, g_2 = 7, g_3 = 3(5) + 7 = 22.

Now dividing P(m) by 49 gives

P(m)/49 = (2401 m^3 - 147 m^2 + 3m) 5^3 - 3(-1 + 49 m )5 + 7

and the question is what happens to the g's, but look now at P(0)/49)
as that is

P(0)/49 = 3(5) + 7

as two factors of 7, each 7, have beeen divided off, which is easy to
see.

But 7, 7, and 49 are NOT functions of m, as they are just numbers, so
those factors must go *independent* of the value of m, which means
that what value of m I choose doesn't matter so now I can go to the
full expression and get

P(m)/49 = (5 a_1/7 + 1)(5 a_2/7 + 1)(5 a_3 + 7)

here it may seem that I just arbitarily divided through, but consider
what happens if you try some other combination, like

P(m)/49 =

(5 a_1/7^{2/3} + 7^{1/3})(5 a_2/7^{2/3} + 7^{1/3})(5 a_3/7^{2/3} +
7^{1/3})

as then letting m=0 gives

P(0)/49 = 7^{1/3} 7^{1/3} (5(3)/7^{2/3} + 7^{1/3}).

While it may seem possible that the 7's roam around based on the value
of m, there's just no mathematical reason for them to do so because,
well, 7 is 7, and it is NOT a function of m.

Now the problem is based on the factorization

P(m)/49 = (5 a_1/7 + 1)(5 a_2/7 + 1)(5 a_3 + 7)

two of the a's *should* have 7 as a factor, and in fact they do, in a
proper ring, but the ring of algebraic integers has problems, so that
for certain values of m, they won't, while maybe (I haven't checked)
for some value of m, they will.

It's that *inconstency* which shows you there's a problem because
mathematics isn't about being wishy-washy, where sometimes something
works and then other times it doesn't.

That error has sat in mathematics, the body of discoveries commonly
called mathematics, for over a *hundred* years.


James Harris

_Fewer_ symbols, surely?

James Harris wrote:

1. First the problematic definition:

Algebraic integers are defined to be roots of monic polynomials with
integer coefficient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where
"monic" refers to the leading coefficient.

My assertion is that the over hundred year old definition excludes
numbers that have to be included to keep from having contradiction
i.e. mathematical inconsistency.

What part of the *definition* is inconsistent? For it to be
inconsistent would suggest that there are no algebraic integers.


2. The important tool I use is a polynomial:

P(m) = 49((2401 m^3 - 147 m^2 + 3m) 5^3 - 3(-1 + 49 m )5 + 7)

The form of the polynomial allows me to factor P(m) into
non-polynomial factors, and the factorization with those factors is

P(m) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7)

where the a's are roots of the following cubic:

a^3 + 3(-1 + 49m)a^2 - f^2(2401 m^3 - 147 m^2 + 3m).

From the factorization I have three factors

g_1 = (5 a_1 + 7), g_2 =(5 a_2 + 7), g_3=(5 a_1 + 7)

but setting m=0, gives me P(0) = 49(3(5) + 7), which fits with the
cubic as at m=0 it gives


a^3 -3a^2 = 0, so a_1 = a_2=0, a_3 = 3,

to show that at m=0, the three factors are

g_1 = 7, g_2 = 7, g_3 = 3(5) + 7 = 22.

Have you looked at P(1) = 2^2 3^2 7^2 7817?
How about P(-1) = 2 7^2 159059?


Now dividing P(m) by 49 gives

P(m)/49 = (2401 m^3 - 147 m^2 + 3m) 5^3 - 3(-1 + 49 m )5 + 7

and the question is what happens to the g's, but look now at P(0)/49)
as that is

P(0)/49 = 3(5) + 7

as two factors of 7, each 7, have beeen divided off, which is easy to
see.

But 7, 7, and 49 are NOT functions of m, as they are just numbers, so
those factors must go *independent* of the value of m, which means
that what value of m I choose doesn't matter so now I can go to the
full expression and get

P(0) is *also* not a function of m. It is just a number, yet you want
to represent it's factorization quite oddly as if it were a function of m.


P(m)/49 = (5 a_1/7 + 1)(5 a_2/7 + 1)(5 a_3 + 7)

here it may seem that I just arbitarily divided through,

It does look that way, yes.

but consider
what happens if you try some other combination, like

P(m)/49 =

(5 a_1/7^{2/3} + 7^{1/3})(5 a_2/7^{2/3} + 7^{1/3})(5 a_3/7^{2/3} +
7^{1/3})

as then letting m=0 gives

P(0)/49 = 7^{1/3} 7^{1/3} (5(3)/7^{2/3} + 7^{1/3}).

While it may seem possible that the 7's roam around based on the value
of m, there's just no mathematical reason for them to do so because,
well, 7 is 7, and it is NOT a function of m.

But P(m) is a function of m, while P(0) is not. Why should they behave
the same way? Have you done any work with other values of m, such as 1?


Now the problem is based on the factorization

P(m)/49 = (5 a_1/7 + 1)(5 a_2/7 + 1)(5 a_3 + 7)

two of the a's *should* have 7 as a factor, and in fact they do, in a
proper ring, but the ring of algebraic integers has problems, so that
for certain values of m, they won't, while maybe (I haven't checked)
for some value of m, they will.

You claim they should, but that would only be the case if P(m) and P(0)
were the same thing. They aren't even the same type of thing, as they
exist in different rings.

Also, what does any of this have to do with the behavior of algebraic
integers as a ring? I don't recall rings having any particular
requirements along the lines of the one you imply here.


It's that *inconstency* which shows you there's a problem because
mathematics isn't about being wishy-washy, where sometimes something
works and then other times it doesn't.

This isn't an inconsistency in the mathematical sense. Your use of
w(m)'s removed the apparent inconsistency quite neatly, except you don't
want to accept it.


That error has sat in mathematics, the body of discoveries commonly
called mathematics, for over a *hundred* years.

What does any of this have to do with the definition of an algebraic
integer? You are talking about properties that are *consequences* of
the definition and that you don't seem to like the way they behave.
Perhaps this says more about your expectations.


--
Will Twentyman
email: wtwentyman at copper dot net

In article (James Harris) writes:
1. First the problematic definition:

Algebraic integers are defined to be roots of monic polynomials with
integer coefficient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where
"monic" refers to the leading coefficient.

My assertion is that the over hundred year old definition excludes
numbers that have to be included to keep from having contradiction
i.e. mathematical inconsistency.

2. The important tool I use is a polynomial:

P(m) = 49((2401 m^3 - 147 m^2 + 3m) 5^3 - 3(-1 + 49 m )5 + 7)

The form of the polynomial allows me to factor P(m) into
non-polynomial factors, and the factorization with those factors is

P(m) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7)

where the a's are roots of the following cubic:

a^3 + 3(-1 + 49m)a^2 - f^2(2401 m^3 - 147 m^2 + 3m).

From the factorization I have three factors

g_1 = (5 a_1 + 7), g_2 =(5 a_2 + 7), g_3=(5 a_1 + 7)

Note that the g's and a's are functions of m.

but setting m=0, gives me P(0) = 49(3(5) + 7), which fits with the
cubic as at m=0 it gives

a^3 -3a^2 = 0, so a_1 = a_2=0, a_3 = 3,

to show that at m=0, the three factors are

So when m=0, 5.a_1(0) + 7 is divisible by 7. Note that this is because
a_1(0) is divisible by 7.

g_1 = 7, g_2 = 7, g_3 = 3(5) + 7 = 22.

Now dividing P(m) by 49 gives

P(m)/49 = (2401 m^3 - 147 m^2 + 3m) 5^3 - 3(-1 + 49 m )5 + 7

and the question is what happens to the g's, but look now at P(0)/49)
as that is

P(0)/49 = 3(5) + 7

as two factors of 7, each 7, have beeen divided off, which is easy to
see.

But 7, 7, and 49 are NOT functions of m, as they are just numbers, so
those factors must go *independent* of the value of m, which means
that what value of m I choose doesn't matter so now I can go to the
full expression and get

This is a wrong assertion. The distribution of the factors of 7 amongst
the factors of the polynomials depends on the divibility of the a's.
And as the a's are dependent on m, so is the divisibility, and so is
the distribution of the factors.

P(m)/49 = (5 a_1/7 + 1)(5 a_2/7 + 1)(5 a_3 + 7)

here it may seem that I just arbitarily divided through, but consider
what happens if you try some other combination, like

P(m)/49 =

(5 a_1/7^{2/3} + 7^{1/3})(5 a_2/7^{2/3} + 7^{1/3})(5 a_3/7^{2/3} +
7^{1/3})

You remain assuming that the factors of 7 distribute similarly for all
cases amongst the factors of the polynomial. This is false.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

In article (James Harris) writes:

Let me try to follow your reasoning. I omit point 1, because it consists
of preliminary comments only, and is just a rehash of things that have
been written too many times already.

I try to follow this with a quadratic, using the same line of reasoning.

2. The important tool I use is a polynomial:

P(m) = 49((2401 m^3 - 147 m^2 + 3m) 5^3 - 3(-1 + 49 m )5 + 7)

I start with:
Q(m) = 7((2.m^2 - 3.m).5^2 - 3.(-1 + m).5 + 7)

The form of the polynomial allows me to factor P(m) into
non-polynomial factors, and the factorization with those factors is

P(m) = (5 a_1 + 7)(5 a_2 + 7)(5 a_3 + 7)

Q(m) = (5.a1 + 7)(5.a2 + 7)

where the a's are roots of the following cubic:

a^3 - 3(-1 + 49m)a^2 - f^2(2401 m^3 - 147 m^2 + 3m).
('f' is a typo for '7'.)

a^2 + 3(-1 + m).a + 7.(2.m^2 - 3.m)


From the factorization I have three factors

g_1 = (5 a_1 + 7), g_2 =(5 a_2 + 7), g_3=(5 a_1 + 7)
(the second 'a_1' is a typo for 'a_3'.)

I get two factors:
g1 = (5a1 + 7) and g2 = (5a2 + 7)

but setting m=0, gives me P(0) = 49(3(5) + 7), which fits with the
cubic as at m=0 it gives

Setting m=0, gives me Q(0) = 7(3.5 + 7), which fits with the cubic as
at m=0 it gives

a^3 -3a^2 = 0, so a_1 = a_2=0, a_3 = 3,

a^2 - 3a = 0, so a1 = 0, and a2 = 3,

to show that at m=0, the three factors are

g_1 = 7, g_2 = 7, g_3 = 3(5) + 7 = 22.

g1 = 7, g2 = 3.5 + 7 = 22.

Now dividing P(m) by 49 gives

P(m)/49 = (2401 m^3 - 147 m^2 + 3m) 5^3 - 3(-1 + 49 m )5 + 7

Dividing Q(m) by 7 gives
Q(m)/7 = (2.m^2 - 3.m).5^2 - 3.(-1 + m).5 + 7

and the question is what happens to the g's, but look now at P(0)/49)
as that is

P(0)/49 = 3(5) + 7

Look at Q(0)/7 as that is
Q(0)/7 = 3.5 + 7

as two factors of 7, each 7, have beeen divided off, which is easy to
see.

As one factor of 7 has been divided off.

But 7, 7, and 49 are NOT functions of m, as they are just numbers, so
those factors must go *independent* of the value of m, which means
that what value of m I choose doesn't matter so now I can go to the
full expression and get

P(m)/49 = (5 a_1/7 + 1)(5 a_2/7 + 1)(5 a_3 + 7)

Q(m)/7 = (5.a1/7 + 1)(5.a3 + 7)

here it may seem that I just arbitarily divided through, but consider
what happens if you try some other combination, like

P(m)/49 =

(5 a_1/7^{2/3} + 7^{1/3})(5 a_2/7^{2/3} + 7^{1/3})(5 a_3/7^{2/3} +
7^{1/3})

Q(m)/7 = (5.a1/sqrt(7) + sqrt(7))(5.a2/sqrt(7) + sqrt(7))

as then letting m=0 gives

P(0)/49 = 7^{1/3} 7^{1/3} (5(3)/7^{2/3} + 7^{1/3}).
[ I note that this evaluates to 5.3 + 7 = 22, the correct answer.]

Q(0)/7 = sqrt(7) (5.3/sqrt(7) + sqrt(7)) = 22.

While it may seem possible that the 7's roam around based on the value
of m, there's just no mathematical reason for them to do so because,
well, 7 is 7, and it is NOT a function of m.

Now the problem is based on the factorization

P(m)/49 = (5 a_1/7 + 1)(5 a_2/7 + 1)(5 a_3 + 7)

Q(m)/7 = (5.a1/7 + 1)(5.a2 + 7)

two of the a's *should* have 7 as a factor, and in fact they do, in a
proper ring, but the ring of algebraic integers has problems, so that
for certain values of m, they won't, while maybe (I haven't checked)
for some value of m, they will.

One of the a's *should* have 7 as a factor, and in fact it does, in a
proper ring, at least that would be the case according to you reasoning.
But....

Remember the quadratic for a:
a^2 + 3(-1 + m).a + 7.(2.m^2 - 3.m)
set m = 1. We get
a^2 - 7
the roots are +- sqrt(7). Which of these two is divisible by 7? You
may note that if you have a ring where one is divisible by 7, the other
is also divisible by 7. So in a ring where one is divisible by 7, 7
is a unit, but in that case both are divisble by 7; in fact all numbers
in that ring are divisible by 7.

Now please state where I am not correct.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

On Mon, 20 Oct 2003 15:46:52 -0400, Will Twentyman
wrote:

James Harris wrote:

1. First the problematic definition:

Algebraic integers are defined to be roots of monic polynomials with
integer coefficient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where
"monic" refers to the leading coefficient.

My assertion is that the over hundred year old definition excludes
numbers that have to be included to keep from having contradiction
i.e. mathematical inconsistency.

What part of the *definition* is inconsistent? For it to be
inconsistent would suggest that there are no algebraic integers.

No, if the definition showed there were no algebraic integers
there would be nothing inconsistent about that (until we found
another proof that there _is_ at least one algebraic integer).

There's simply no way that this _definition_ can possibly be
inconsistent. As always James is not saying exactly what he
means - he means that the definition, together with other
facts that he imagines are true, is inconsistent.


************************

David C. Ullrich

David C. Ullrich wrote:
On Mon, 20 Oct 2003 15:46:52 -0400, Will Twentyman
wrote:


James Harris wrote:


1. First the problematic definition:

Algebraic integers are defined to be roots of monic polynomials with
integer coefficient e.g. x^3 + 3x + 1 or x^234 - 34x^12 + 17, where
"monic" refers to the leading coefficient.

My assertion is that the over hundred year old definition excludes
numbers that have to be included to keep from having contradiction
i.e. mathematical inconsistency.

What part of the *definition* is inconsistent? For it to be
inconsistent would suggest that there are no algebraic integers.


No, if the definition showed there were no algebraic integers
there would be nothing inconsistent about that (until we found
another proof that there _is_ at least one algebraic integer).

True. Of course, the fact that x+2 has solution -2 trivially shows that
there is at least one algebraic integer.

There's simply no way that this _definition_ can possibly be
inconsistent. As always James is not saying exactly what he
means - he means that the definition, together with other
facts that he imagines are true, is inconsistent.

Unfortunately, James has yet to take a class in which he finds out how
mathematicians move from definition to properties. He seems to be
convinced that we decide on the properties, and then construct the
definition to fit. While that happens sometimes, as often we start with
the definition and *then* explore the properties.

--
Will Twentyman
email: wtwentyman at copper dot net