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 Post subject: Irreducible polynomialsPosted: Tue, 6 Dec 2011 00:41:36 UTC
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Joined: Mon, 28 Nov 2011 22:48:48 UTC
Posts: 10
Problems statement:

(i) Give an example of a polynomial f(X) ∈ R[X] which is irreducible in R[X] but
not in C[X].
(ii) Are there polynomials in R[X] which are irreducible in C[X] but not in R[X]?
(iii) Which of the following polynomials are irreducible in R[X]:
f(X) = X^3+ 5X − 2 , g(X) = X^2 + 3 , h(X) = X^4 − 1 ?
(iv) Which of the following polynomials are irreducible in Q[X]:
f(X) = X^3 − 2 , g(X) = X , h(X) = X^2+ 2X + 1

I think the polynomial X^2 + 1 is a good example for this question

I don't think so, because C[X] contains R[X]; therefore, if the polynomial was irreducible in C[X] then it should be irreducible in R[X]

f(x) // I couldn't factor/reduce it, but maybe it can be reduced?
h(x) can be reduced to (x^2 + 1)(x+1)(x-1) does this mean it is reducible or is it irreducible in R[X] because one can't reduce (x^2 + 1)?

f(x) is reducible -- difference of cubes? where 2 = (2^1/3)^3
g(x) is irreducible -- it is linear.
h(x) is reducible -- (x+1)^2 = (x+1)(x+1)

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 Post subject: Re: Irreducible polynomialsPosted: Tue, 6 Dec 2011 01:50:53 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
Gantz wrote:
Problems statement:

(i) Give an example of a polynomial f(X) ∈ R[X] which is irreducible in R[X] but
not in C[X].
(ii) Are there polynomials in R[X] which are irreducible in C[X] but not in R[X]?
(iii) Which of the following polynomials are irreducible in R[X]:
f(X) = X^3+ 5X − 2 , g(X) = X^2 + 3 , h(X) = X^4 − 1 ?
(iv) Which of the following polynomials are irreducible in Q[X]:
f(X) = X^3 − 2 , g(X) = X , h(X) = X^2+ 2X + 1

I think the polynomial X^2 + 1 is a good example for this question

I don't think so, because C[X] contains R[X]; therefore, if the polynomial was irreducible in C[X] then it should be irreducible in R[X]

f(x) // I couldn't factor/reduce it, but maybe it can be reduced?
h(x) can be reduced to (x^2 + 1)(x+1)(x-1) does this mean it is reducible or is it irreducible in R[X] because one can't reduce (x^2 + 1)?

f(x) is reducible -- difference of cubes? where 2 = (2^1/3)^3
g(x) is irreducible -- it is linear.
h(x) is reducible -- (x+1)^2 = (x+1)(x+1)

For (i), you are fine.

For (ii), you're right but your argument is a bit flaccid. By definition of irreducible, there are no polynomials with complex coefficients such that the polynomial factors as the product of those, but real polynomials are in particular complex polynomials (as you said), so a factorization there would also be a complex factorization, producing a contradiction.

For (iii) Recall that the only irreducibles over are linear polynoimals, and quadratics with an imaginary root (because is algebraically closed and any algebraic extension of is contained in its algebraic closure. But then you can compare degrees and get a contradiction if any of the degree 3 or higher were irreducible.)

For (iv) Remember you're working over and not , so the factor is not a rational polynomial.

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