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Math Help - Polynomials written as a product of irreducible polynomials

  1. #1
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    Polynomials written as a product of irreducible polynomials

    Can someone please help me with this question:



    Prove by induction that every polynomial in R[x] can be written as a product of irreducible polynomials in R[x] (Note: R is the real numbers)




    Thank you!
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  2. #2
    Senior Member Shanks's Avatar
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    Hint: prove by induction on the degree of polynomial in R[x].
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  3. #3
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    Quote Originally Posted by Shanks View Post
    Hint: prove by induction on the degree of polynomial in R[x].
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    Quote Originally Posted by pseudonym View Post
    Can someone please help me with this question:



    Prove by induction that every polynomial in R[x] can be written as a product of irreducible polynomials in R[x] (Note: R is the real numbers)




    Thank you!
    Here's a hint:

    Base case: for n=1 every polynomial of degree 1 is irreducible so it can be written as itself.

    Inductive step: We can assume the statement is true for polynomials with degree 1,...,n. You need to show that it is true for polynomials with degree n+1.

    Can you continue from here?
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    Quote Originally Posted by Roam View Post
    Here's a hint:

    Base case: for n=1 every polynomial of degree 1 is irreducible so it can be written as itself.

    Inductive step: We can assume the statement is true for polynomials with degree 1,...,n. You need to show that it is true for polynomials with degree n+1.

    Can you continue from here?

    Two hints more (well read and better thought, the following hints are about 90% of the answer to your question):

    1) Any real polynomial of odd degree has a real root (m.v.t. for continuous functions) , and:

    2) If z\in\mathbb{C} is a root of a real polynomial, then also \overline{z} is , or in other words: complex non-real roots of a real polynomial come in conjugate pairs.

    Tonio
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  6. #6
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by tonio View Post
    Two hints more (well read and better thought, the following hints are about 90% of the answer to your question):

    1) Any real polynomial of odd degree has a real root (m.v.t. for continuous functions) , and:

    2) If z\in\mathbb{C} is a root of a real polynomial, then also \overline{z} is , or in other words: complex non-real roots of a real polynomial come in conjugate pairs.

    Tonio
    Are these hints necessary to the proof? I mean, you start the induction (strongly!) and end up with two cases. These two cases easily break down by your inductive argument. Nothing too fancy is needed.

    Unless you are hinting towards a non-inductive proof?
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  7. #7
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    To prove the inductive step you have to note that all polynomials of degree n+1 are either irreducible (if so they can be written as a themselves), or written as a product of two polynomials with smaller degree. Then you can use the assumption and write down each of them as a product of irreducible polynomials.
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  8. #8
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    Quote Originally Posted by Swlabr View Post
    Are these hints necessary to the proof? I mean, you start the induction (strongly!) and end up with two cases. These two cases easily break down by your inductive argument. Nothing too fancy is needed.

    Unless you are hinting towards a non-inductive proof?

    As many other times in the past I misread, or better: didn't read completely. The induction proof works for polynomials over any integral domain, and since I read "real" I assumed (because, and I checked, I did NOT read it in the OP) that he meant : any real polynomial can be written as the product of irreducible polynomials....of degree 1 and/or 2 .

    My hints, thus, as huge overkill.

    Tonio
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