Relatively Prime Polynomials in Extension Fields

[e(ho0n3]

Homework Statement
Recall that two polynomials f(x) and g(x) from F[x] are said to be relatively prime if there is no polynomial of positive degree in F[x] that divides both f(x) and g(x). Show that if f(x) and g(x) are relatively prime in F[x], they are relatively prime in K[x], where K is an extension of F.

The attempt at a solution
I'm guessing this will be a proof by contradiction where the contradiction will be that f(x) and g(x) are not relatively prime in F[x]:

Let p(x) be an irreducible factor of the divisor of f(x) and g(x) in K[x]. How can I show that p(x) divides f(x) and g(x) in F[x]?

 

[n_bourbaki]

You can't do what you ask for in your last sentence: p(x) isn't even necessarily in F[x]. But the question doesn't ask you to show that.

 

[e(ho0n3]

Is there another way of getting a contradiction? I can't think of anything else.

 

[n_bourbaki]

Why have you stopped what you're doing? The only thing wrong was your assumption that any factor p(x) in K[x] must be in F[x].

Consider an example if it helps you. E.g. x^2+1 and x^4-1 over Q[x]. These are not relatively prime, but you don't prove that by considering *one* irreducible linear factor in C[x], since x-i and x+i are not in Q[x].

 

[e(ho0n3]

I got the impression that my whole approach was wrong.

So are you saying I have to consider all the irreducible factors in K[x]? In what way? Perhaps the product of some of the irreducible linear factors of the divisor is a member of F[x], e.g. (x - i)(x + i) = x² + 1 in your example?

 

[n_bourbaki]

Let's suppose that I have a polynomial f(x) with rational coefficients. And suppose that, passing to the complex numbers, I know x-i is a root of f. What other linear factor must be a root of f(x)?

What do you know about things like Aut(K:F)? The field automorphisms of K that fix F. Such as complex conjugation in Aut(C:Q).

 

[e(ho0n3]

Let's suppose that I have a polynomial f(x) with rational coefficients. And suppose that, passing to the complex numbers, I know x-i is a root of f. What other linear factor must be a root of f(x)?

I'm guessing x + i. I don't really know. To find another linear factor I would try to find a zero of f(x)/(x - i).

What do you know about things like Aut(K:F)? The field automorphisms of K that fix F. Such as complex conjugation in Aut(C:Q).

I've never seen that notation before.

My problem is from a chapter on extension and splitting fields.

 

[n_bourbaki]

Yes, that makes sense - you can assume that K is a splitting field for f(x).

You do know the first part: complex roots of a real polynomial occur in complex conjugate pairs.

 

[e(ho0n3]

You do know the first part: complex roots of a real polynomial occur in complex conjugate pairs.

That's an interesting fact. So in the case of Q and C, the contradiction works. But in general? Does p(x) have a conjugate?

 

[e(ho0n3]

I was told that since f(x) and g(x) are relatively prime in F[x], then there are s(x) and t(x) in F[x] with f(x)s(x) + g(x)t(x) = 1. This equation is also valid in K[x] and this supposedly implies that f(x) and g(x) are relatively prime in K[x]. I don't understand this implication. Why is this?

 

[n_bourbaki]

It's just euclid's algorithm. Just think of Z rather than a polynomial algebra.

You can do it with a contradiction for any extension. But you don't need to.

 

[e(ho0n3]

Does f(x)s(x) + g(x)t(x) = 1 imply that f(x) and g(x) are relatively prime? Isn't it the other way around?

 

[n_bourbaki]

It is exactly the same as it is for the integers. You should be able to answer your own question.