Blocking algebra with algebraic integers

In the ring of algebraic integers the expression:

P(x) = (g₁(x) + 1)(g₂(x) + 2)

is blocked if P(x) is a primitive quadratic irreducible over Q, and g₁(0) = g₂(0) = 0, but g₁(x) does not equal 0 for all x, for certain solutions.

(Updating October 26, 2017: Original argument was in error as assumes c₁ is a non-unit integer.)

Proof:

Let P(x) = (g₁(x) + 1)(g₂(x) + 2)

where P(x) is a quadratic irreducible over Q, g₁(0) = g₂(0) = 0, but g₁(x) does not equal 0 for all x.

Introduce new functions f₁(x), and f₂(x), and non-zero integers c₁ and c₂, where I'll use the second one first, as let:

g₂(x) = f₂(x) + c₂

where c₁ = c₂ + 2

Now make the substitution:

P(x) = (g₁(x) + 1)(f₂(x) + c₂ +  2) = (g₁(x) + 1)(f₂(x) + c₁)

Now multiply both sides by c₁,

c₁*P(x) =  (c₁g₁(x) + c₁)(f₂(x) + c₁)

And now, let g₁(x) = f₁(x)/c₁, and make that substitution to get:

c₁*P(x) =  (f₁(x) + c₁)(f₂(x) + c₁)

Indistinguishable, the f's must be roots of the same polynomial function, as symmetry has been forced.

But now the requirement that g₁(x) = f₁(x)/c₁, means it is blocked in the ring of algebraic integers if c₁ is not 1 or -1, unless f₁(x) is itself a polynomial function. But if P(x) is irreducible over Q, then that is not possible; therefore, g₁(x) cannot be an algebraic integer function.

Proof complete.

Here's an example of the blocking with actual equations that fulfill all of the restrictions.

Let c₂ = 5, f₁(x) = 5a₁(x), and f₂(x) = 5a₂(x), where the a's are roots of

a² - (7x-1)a + (49x² - 14x) = 0

so the f's are algebraic integer functions, since the polynomial expression is monic, as long as x is an algebraic integer. That then requires that

P(x) = 175x² - 15x + 2

which is not reducible over Q.

Notice you can see the consequences if you let x = 1, then a² - 6a + 35 = 0, which would imply that just one of the roots has 7 as a factor, which is NOT possible in the ring of algebraic integers, as in that ring neither can have 7 as a factor.

So it is proven that algebra that is allowed for polynomial factorizations is blocked within the ring of algebraic integers for non-polynomial factorizations.


James Harris