Factorising Polynomials in Z[x]

[pivoxa15]

Homework Statement

How do you factorise polynomials in Z[x]? Z is not a field so you can't use the theorem 'a is a root of f <=> (x-a) divides f'

The Attempt at a Solution

Would you map the polynomials from Z[x] to Q[x] by multiplying by 1, since all elements in Z[x] are in Q[x]. Then factorise in Q[x] usiong the theorem above. If the factors have coefficients in Z than map (multiply by 1) these back to Z[x] so you have factorised these polynomials and they exist in Z[x]. If the factors in Q[x] have coefficients not in Z then you can't map back to Z[x] hence these are not factorisable in Z[x].
Is this how you would factorise polynomials in Z[x], formally?

 

[StatusX]

Look up Gauss' lemma.

 

[pivoxa15]

How does Gauss' lemma help?

 

[matt grime]

What makes you think passing to Q[x] suddenly makes things easier? What is the factorization ox x^2-2 in Z[x]? And in Q[x]? The x-a a root thing is still as valid/not valid in Z[x] as in Q[x]. In fact that is what (one of) Gauss's Lemmas states: a poly is irreducible over Z iff it is irreducible over Q.

You factorise it 'by doing it'. I.e. trying to find a factorization. In general this is *very hard* and can't be done by any decent analytic means. It is however possible to show there are no integral roots by exhaustion.

 

[pivoxa15]

I like to pass over to Q[x] so I can use the theorem I quoted in the OP because Q is a field. I like to justify every step I take instead of relying on intuitition.

The version you quoted isn't in here
http://en.wikipedia.org/wiki/Gauss's_lemma_(polynomial)

 

[matt grime]

Really? Second bullet point. If a poly is irreducible over Z it is irreducible over Q. (Obviously if it is reducible over Z it is reducible over Q).

 

[pivoxa15]

Right. But the second point only has one way from the integers to the rationals. You had iff in your previous post?! Consider f(x)=2x^2+2x+2 is irreducible in Q[x] but is reducible in Z[x] i.e f(x)=2(x^2+x+1).

Anyhow, if you follow my method, and map the polynomials to Q[x] then you can (potentially) factorise by finding zeros of it (which is allowed by the theorem). Then follow two implications,
Let f(x) by original polynomial in Z[x]
1. f(x) primitive in Z[x] and irreducible in Q[x] => irreducible in Z[x]
2. f(x) is monic and reducible in Q[x] => reducible in Z[x]

It was convienent that all f(x) I was interested in factorising were monic hence also primitive so by mapping them to Q[x] I was able to tell if it was irreducible or reducible in Z[x], rigorously. If the word is appropriate here.

But would you say 'mapping' is a good word here? I am merely factoring specific polynomials given to me which were said to be in Z[x]. But I chose to test reducibility in Q[x]. Is a map really needed here? If not than what should I say or justify me taking the polynomials to Q[x] and doing the maths?

 

[matt grime]

I pointed out the 'other' direction. It is trivial: irreducible over Q implies irreducible over Z. That really is trivial.

 

[pivoxa15]

But what about

f(x)=2x^2+2x+2 is irreducible in Q[x] but is reducible in Z[x] i.e f(x)=2(x^2+x+1) with neither 2 nor x^2+x+1 is a unit in Z[x]

So you need
f(x) primitive in Z[x] and irreducible in Q[x] => irreducible in Z[x]
Correct?

The definition of irreducible I use is, P is irreducible <=>
P is non unit and if p=ab than a is a unit or b is a unit
What about my mapping question in my previous post? Is it appropriate?

 

[matt grime]

Ok, modulo trivialities like dividing by a constant, jeez are you missing the point.