Thursday, September 16, 2010
JSH: More units
Though it took YEARS for me to figure out, for the big picture people trying to get a handle on what could be wrong with the ring of algebraic integers, you can try focusing on one key expression:
x^2 + bx + 2 = (x + 2u_1)(x + u_2)
where 'b' is a non-zero integer chosen such that the roots of the quadratic are non-rational, as, guess what?
Then u_1 cannot exist in the ring of algebraic integers, and u_2 cannot be a unit within that ring.
If you understand the above and figure out why then you should have a good handle on the problem and can begin surveying the scope of destruction this result puts across number theory over the last hundred years.
Sorry, had to toss in some hyperbole there—I'm so good at it!
Clue: Major issue is with defining algebraic integers as roots of monic polynomials with integer coefficients. With the expression above, consider solving for u_1. You will get a non-monic!!!
It DID take me years to figure it out, and I've been working at explaining it for more years. I've also fixed the problem—closing what I call now a coverage gap—with a ring I call the object ring.
(A perk of being the discoverer is getting to name things. Really cool perk.)
Google: object ring
Think about that fix and how I solved the problem and you'll see the rational behind that definition which includes the ring of algebraic integers, and the units that ring leaves out.
Try that clue above if you're puzzling over it greatly. The problem has existed for over a hundred years. Seems reasonable to suppose that you may take a little while to really figure it out.
x^2 + bx + 2 = (x + 2u_1)(x + u_2)
where 'b' is a non-zero integer chosen such that the roots of the quadratic are non-rational, as, guess what?
Then u_1 cannot exist in the ring of algebraic integers, and u_2 cannot be a unit within that ring.
If you understand the above and figure out why then you should have a good handle on the problem and can begin surveying the scope of destruction this result puts across number theory over the last hundred years.
Sorry, had to toss in some hyperbole there—I'm so good at it!
Clue: Major issue is with defining algebraic integers as roots of monic polynomials with integer coefficients. With the expression above, consider solving for u_1. You will get a non-monic!!!
It DID take me years to figure it out, and I've been working at explaining it for more years. I've also fixed the problem—closing what I call now a coverage gap—with a ring I call the object ring.
(A perk of being the discoverer is getting to name things. Really cool perk.)
Google: object ring
Think about that fix and how I solved the problem and you'll see the rational behind that definition which includes the ring of algebraic integers, and the units that ring leaves out.
Try that clue above if you're puzzling over it greatly. The problem has existed for over a hundred years. Seems reasonable to suppose that you may take a little while to really figure it out.
# posted by James Harris @ 22:40 
