One of my favorite demonstrations uses a seemingly simple construction:
P(x) = (g1(x) + 1)(g2(x) + 2)
where P(x) is a quadratic with integer coefficients, g1(0) = g2(0) = 0, but g1(x) does not equal 0 for all x.
Turns out that expression can be pushed out of the ring of algebraic integers.
For example, like have g1(x) = x, and g2(x) = x, then, of course P(x) = x2 + 3x + 2.
But I pushed the envelope by imagining the quadratic is what's called primitive and irreducible over Q, which means that it does NOT factor into polynomials, where an additional factor can force outside the ring of algebraic integers.
And have shown where I multiplied by 7, but this time I will multiply by 11,which is the additional factor:
Introduce new functions f1(x), and f2(x), where I'll use the second one first, as let
g2(x) = f2(x) + 9.
Now make the substitution:
P(x) = (g1(x) + 1)(f2(x) + 9 + 2) = (g1(x) + 1)(f2(x) + 11)
Multiply both sides by 11:
11*P(x) = (11g1(x) + 11)(f2(x) + 11)
Now, let g1(x) = f1(x)/11, and make that substitution to get:
11*P(x) = (f1(x) + 11)(f2(x) + 11)
Now multiply it out:
11*P(x) = f1(x)*f2(x) + 11(f1(x) + f2(x)) + 121
And introduce H(x), where I like using the capital letter here for visual reasons, but it is not to signify H(x) must be a polynomial, where:
f1(x) + f2(x) = H(x), so: f2(x) = -f1(x) + H(x),
and make that substitution, to get:
11*P(x) = f1(x)*(-f1(x) + H(x)) + 11H(x) + 121
So you have:
11*P(x) = -f12(x) + H(x)f1(x) + 11H(x) + 121
f12(x) - H(x)f1(x) - 11H(x) - 121 + 11*P(x) = 0
And you can solve for f1(x) using the quadratic formula.
Notice you can see the ambiguity between the f's right there, which prevents you from knowing which one has 11 as a factor, as of course looks the same with f2.
And if H(x) is in the ring of algebraic integers, since the coefficients of P(x) are integers, f1(x) must be in the ring of algebraic integers.
Notice you can start with:
P(x) = (g1(x) + 1)(g2(x) + 2)
and multiply by 13 if you wish, as of course it is bare. What you multiply with is human choice. I've chosen 7 and 11, for my own reasons. And you may now wonder, so how do you pick H(x)?
Does it matter?
One way of looking at it, going in this direction, you can choose H(x) and in that way force the g's, which means it is a handle on the original factorization.
Your choice of H(x) tells the math how the original factorization must go.
Are there any circumstances under which the f's that result from your choice could NOT be in the ring of algebraic integers?
Well the only way is if you deliberately make H(x) not be in the ring of algebraic integers by picking coefficients outside of it, but even if you try I think it's kind of hard to do and have all the equations work. For instance since f1(0) = 0, and f2(0) = -9, you know it must be that H(0) = -9.
Actually I don't know if you can force H(x) to not be an algebraic integer and have all the equations work. But even if you can, you'd have to be an ornery (and clever) person who just made that happen.
With my own research H(x) has been linear with integer coefficients.
Those original g's are weird then, eh? Solutions for algebraic integer x cannot all be seen, as they are not both expressible as algebraic integers, so can't be seen directly, but you can multiply the one that is not by an integer and see the result!
And they're not too picky about the integer either. It just needs to be nonzero and not 1 or -1, and they will appear, as the result is an algebraic integer.
I came up with the object ring to categorize these peculiar numbers, and call them and integers, objects. So you can say mathematically that given an object o1, which is a solution to one of the g's from my original factorization which is not an algebraic integer, it is visible with 7o1 or 11o1 with the examples I've used. It doesn't care.
Oh yeah, so what about o2? Well it is an algebraic integer, but which is which? No way to tell. But, yup, o1o2 is an algebraic integer.
Non algebraic integer objects are fascinating critters!
Their refusal to be seen directly is NOT intuitive.
These non-rational objects which are not algebraic integers are an undiscovered country of number theory. If you don't believe they exist and are being befuddled by the idea of fractions then how can I multiply the same number by 7 or 11 to get an algebraic integer? Would it help if I did it again with 9? Or 16?