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Sunday, August 05, 2007

Wrapper theorem and ring of algebraic integers

The ring of algebraic integers has crucial properties which are unique to it involving the use of what I call wrapper values to allow certain factorizations to exist in that ring.

In an integral domain let

d1d2P(x) = (f1(x) + d1c1)(f2(x) + d2c2)

where the d's and c's are non-zero integers, where the d's are coprime to the c's and c1 is coprime to c2, P(x) is a non-monic primitive polynomial with integer coefficients, and where f1(0) = f2(0) = 0 so P(0) is coprime to d1 and d2, as then P(0) = c1c2.

If the ring is the ring of algebraic integers, with non-zero integer x, and d1 = p1 and d2 = p2 where p1 and p2 are differing prime numbers, and f1(x) and f2(x) are non-rational there cannot exist g1(x) and g2(x) such that

d1d2P(x) = (d1g1(x) + d1c1)(d2g2(x) + d2c2)

where

d1g1(x) = f1(x) and d2g2(x) = f2(x).

Proof:

Since P(x) is a polynomial with integer coefficients f1(x) and f2(x) must have values that are roots of the same polynomial. But if f1(x) has d1 as a factor and f2(x) does not, then you can find a non-monic primitive polynomial with integer coefficients irreducible over rationals, which contradicts a well established elementary theorem in number theory that none exists.

So then, how can you divide off d1 and d2?

One way is for the ring to employ what I call wrappers.

Consider w1 and w2, where

d1d2P(x) = ((w1d1)(w1(g1(x)+c1-d2) + (w1)2d1d2)((w2d2)(w2(g2(x)+c2-d1) + (w2)2d1d2)

where w1d1 and w2d2 are roots of a monic polynomial with integer coefficients, so those products are in the ring of algebraic integers while the factors are not.

In my ring of objects the wrappers are units and w1w2 = 1 or -1, while they are NOT units in the ring of algebraic integers.

So the ring of algebraic integers can multiply the factors (f1(x) + d1) and (f2(x) + d2) by numbers that are units in the object ring while also shifting them internally so that it is possible to divide off the d's.

What is left is then

P(x) = ((w1(g1(x)+c1-d2) + w1d2)((w2(g2(x)+c2-d1) + w2d1)

where that detail is available in the object ring as the individual elements are not all available in the ring of algebraic integers.


James Harris
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