## 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.)