Friday, October 20, 2006
JSH: Simpler demonstration
Turns out you can use some numbers that are special in that they are coprime to each other with respect to some prime:
5 + (sqrt(37) - sqrt(13))/2 and (sqrt(37) - sqrt(13))/2 are coprime to each other with respect to 3 which can be seen by subtracting one from the other, but each has its own factors in common with 3.
Therefore, 5 + (sqrt(37) - sqrt(13))/2 must share the same factors in common with 3 as sqrt(37) + sqrt(13) as that is coprime to sqrt(37) - sqrt(13).
Also though (sqrt(37) - sqrt(13))*(sqrt(37) + sqrt(13)) = 24, so there are factors in common with 2 in there, so now I know that letting
z = 2(10 + sqrt(37) - sqrt(13))/(sqrt(37) + sqrt(13))
should balance those out, and now I find that z is a root of
3z4 - 50z3 + 74z2 + 3100z + 248 = 0
which is a non-monic polynomial irreducible over rationals proving that z is NOT an algebraic integer.
Notice that this demonstration is easy—you just need to find a situation where you have two numbers coprime to some other number with respect to a prime.
It also didn't require a special tool I'd think to find such numbers though I did use the one I've brought up in other posts. But I do think it takes a will to NOT ever find such simple examples to hold on to flawed theories.
So why must this refute standard usage of Galois Theory and standard ideas about the ring of algebraic integers?
Because by proving that two numbers share the same factors in common with 3 by showing they are coprime to the same number, I show that when other factors are handled—in this case 2—one should divide into the other, and they do, in a proper ring without the coverage problem of the ring of algebraic integers.
And remember I have proven a problem with that ring before!!! The paper I wrote shows it, as do numerous postings where I've explained the mathematics in such detail that I bring it down to the distributive property.
There has been a will to remain wrong as people have fought this result.
5 + (sqrt(37) - sqrt(13))/2 and (sqrt(37) - sqrt(13))/2 are coprime to each other with respect to 3 which can be seen by subtracting one from the other, but each has its own factors in common with 3.
Therefore, 5 + (sqrt(37) - sqrt(13))/2 must share the same factors in common with 3 as sqrt(37) + sqrt(13) as that is coprime to sqrt(37) - sqrt(13).
Also though (sqrt(37) - sqrt(13))*(sqrt(37) + sqrt(13)) = 24, so there are factors in common with 2 in there, so now I know that letting
z = 2(10 + sqrt(37) - sqrt(13))/(sqrt(37) + sqrt(13))
should balance those out, and now I find that z is a root of
3z4 - 50z3 + 74z2 + 3100z + 248 = 0
which is a non-monic polynomial irreducible over rationals proving that z is NOT an algebraic integer.
Notice that this demonstration is easy—you just need to find a situation where you have two numbers coprime to some other number with respect to a prime.
It also didn't require a special tool I'd think to find such numbers though I did use the one I've brought up in other posts. But I do think it takes a will to NOT ever find such simple examples to hold on to flawed theories.
So why must this refute standard usage of Galois Theory and standard ideas about the ring of algebraic integers?
Because by proving that two numbers share the same factors in common with 3 by showing they are coprime to the same number, I show that when other factors are handled—in this case 2—one should divide into the other, and they do, in a proper ring without the coverage problem of the ring of algebraic integers.
And remember I have proven a problem with that ring before!!! The paper I wrote shows it, as do numerous postings where I've explained the mathematics in such detail that I bring it down to the distributive property.
There has been a will to remain wrong as people have fought this result.
# posted by James Harris @ 00:31 
