Friday, December 19, 2008
JSH: Questioning Galois Theory
One of the more remarkable things I have is a weirdly simple result on the complex plane which brings into question Galois Theory, which is just such a huge thing that it's hard to surmount disbelief despite the simplicity of the proof.
The way it works is I figured out this inventive way to factor a polynomial in the complex plane:
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.
It may seem simple enough but it's complexities have sparked arguments for YEARS as in literally years on math newsgroups, so I want to step back a bit and show you another more traditional factorization:
7(x^2 + 3x + 2) = (7x + 7)(x+2)
which is trivial, but is actually in all key respects the same as the factorization above!!!
That is where the issues start as, of course, people disagreeing with this result would attack that claim and besides, you can see above that there are two 7's on the right hand side, not one, with my weird, inventive factorization.
But, normalize and you get a different picture as using a_1(x) = b_1(x), and a_2(x) = b_2(x) - 1, you get
7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)
where b_1(0) = 0, b_2(0) = 0, which looks a lot more like
7(x^2 + 3x + 2) = (7x + 7)(x+2)
except, wait a minute, the 7 is obscured by the function b_1(x), while it is visible with the linear function, and that's it.
So yes, essentially they are the SAME in terms of you have 7 multiplied times one factor of a factorization, where in one case that 7 is obscured by a function, which is non-linear, while it is visible with the second case where you have a linear function.
So in a way, all the arguing over the years has been an issue about non-linear versus linear functions, as by the distributive property:
a*(f(x) + b) = a*f(x) + a*b
doesn't care about the TYPE of function f(x). But if you accept all of the above there is no argument, you now have that with
a^2 - (7x-1)a + (49x^2 - 14x) = 0
one of the roots should always have 7 as a factor as only one of the functions actually is multiplied by 7, and that's where things get dicey for Galois Theory.
Too big of a leap? Well let's step back a bit and stick in some numbers!
Let x=1, then you have a^2 - 6a + 35 = 0, which you can solve using the quadratic formula to get
a = (6 +/- sqrt(-104))/2 = 3 +/- sqrt(-26)
and I don't see 7 in there at all! It's obscured by the square root. Here's an example of that with easy numbers:
a^2 + 4a + 3 = 0, so
a = (-4 +/- sqrt(4))/2
and I don't see a factor of 3 in there—unless I resolve the square root.
But the problem with 3 +/- sqrt(-26), is that you CANNOT resolve the square root, so you CANNOT see the 7, which we just proved on the complex plane MUST be there.
So you have mathematical proof as an instrument, like for a pilot flying at night—you cannot see the 7 directly but you proved it's there on the complex plane so you know logically it HAS to be there—so in a way you're flying at night, like a pilot, needing to rely on his mathematical instruments.
But here's where things get really messy, as mathematicians well-trained and taught in over a hundred years of number theory will tell you, if you tell them there's a 7 in there, that you are wrong, and they will tell you they can prove you are wrong!!!
So what gives? I just stepped through a simple argument showing how it was just linear functions versus non-linear where just being non-linear doesn't change the distributive property so we know a 7 is there, but now I'm telling you that the experts in the field of mathematics will tell you it's not there, and say they mathematically prove it's not there, and oh yeah, what does this
have to do with Galois Theory anyway?
Well, if you understand how the experts in the mathematical field are wrong—like, hey, you can trivially prove on the complex plane they are wrong—and understand WHY they are wrong, you come across the problem that they built Galois Theory on those reasons why they are wrong!
Now they are fully invested in their beliefs!!!
You're talking about prizewinning theorists and top-ranked academics who work in highly prestigious institutions with all kinds of authority and social power to tell you that no matter how well you THINK you know the complex field, or how logical the argument above about linear functions are not that different from non-linear ones when you multiply them by 7 sounds to you, it's all crap because they're the experts and they have a hundred years of edifice built on it being crap.
So they will tell you you are wrong.
The way it works is I figured out this inventive way to factor a polynomial in the complex plane:
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.
It may seem simple enough but it's complexities have sparked arguments for YEARS as in literally years on math newsgroups, so I want to step back a bit and show you another more traditional factorization:
7(x^2 + 3x + 2) = (7x + 7)(x+2)
which is trivial, but is actually in all key respects the same as the factorization above!!!
That is where the issues start as, of course, people disagreeing with this result would attack that claim and besides, you can see above that there are two 7's on the right hand side, not one, with my weird, inventive factorization.
But, normalize and you get a different picture as using a_1(x) = b_1(x), and a_2(x) = b_2(x) - 1, you get
7*(175x^2 - 15x + 2) = (5b_1(x) + 7)(5b_2(x)+ 2)
where b_1(0) = 0, b_2(0) = 0, which looks a lot more like
7(x^2 + 3x + 2) = (7x + 7)(x+2)
except, wait a minute, the 7 is obscured by the function b_1(x), while it is visible with the linear function, and that's it.
So yes, essentially they are the SAME in terms of you have 7 multiplied times one factor of a factorization, where in one case that 7 is obscured by a function, which is non-linear, while it is visible with the second case where you have a linear function.
So in a way, all the arguing over the years has been an issue about non-linear versus linear functions, as by the distributive property:
a*(f(x) + b) = a*f(x) + a*b
doesn't care about the TYPE of function f(x). But if you accept all of the above there is no argument, you now have that with
a^2 - (7x-1)a + (49x^2 - 14x) = 0
one of the roots should always have 7 as a factor as only one of the functions actually is multiplied by 7, and that's where things get dicey for Galois Theory.
Too big of a leap? Well let's step back a bit and stick in some numbers!
Let x=1, then you have a^2 - 6a + 35 = 0, which you can solve using the quadratic formula to get
a = (6 +/- sqrt(-104))/2 = 3 +/- sqrt(-26)
and I don't see 7 in there at all! It's obscured by the square root. Here's an example of that with easy numbers:
a^2 + 4a + 3 = 0, so
a = (-4 +/- sqrt(4))/2
and I don't see a factor of 3 in there—unless I resolve the square root.
But the problem with 3 +/- sqrt(-26), is that you CANNOT resolve the square root, so you CANNOT see the 7, which we just proved on the complex plane MUST be there.
So you have mathematical proof as an instrument, like for a pilot flying at night—you cannot see the 7 directly but you proved it's there on the complex plane so you know logically it HAS to be there—so in a way you're flying at night, like a pilot, needing to rely on his mathematical instruments.
But here's where things get really messy, as mathematicians well-trained and taught in over a hundred years of number theory will tell you, if you tell them there's a 7 in there, that you are wrong, and they will tell you they can prove you are wrong!!!
So what gives? I just stepped through a simple argument showing how it was just linear functions versus non-linear where just being non-linear doesn't change the distributive property so we know a 7 is there, but now I'm telling you that the experts in the field of mathematics will tell you it's not there, and say they mathematically prove it's not there, and oh yeah, what does this
have to do with Galois Theory anyway?
Well, if you understand how the experts in the mathematical field are wrong—like, hey, you can trivially prove on the complex plane they are wrong—and understand WHY they are wrong, you come across the problem that they built Galois Theory on those reasons why they are wrong!
Now they are fully invested in their beliefs!!!
You're talking about prizewinning theorists and top-ranked academics who work in highly prestigious institutions with all kinds of authority and social power to tell you that no matter how well you THINK you know the complex field, or how logical the argument above about linear functions are not that different from non-linear ones when you multiply them by 7 sounds to you, it's all crap because they're the experts and they have a hundred years of edifice built on it being crap.
So they will tell you you are wrong.
# posted by James Harris @ 21:29 
