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Math Help - Irreducible polynomial

  1. #1
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    Irreducible polynomial

    Hi everybody,

    Could You help me with the following statement:

    Proof that there are (one variable) polynomials of any natural degree with rational coefficients, which are irreducible over the field of rational numbers.

    Thanks.
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  2. #2
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    It's a direct consequence from Eisenstein's criterion.
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  3. #3
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    Quote Originally Posted by Migotek84 View Post
    Hi everybody,

    Could You help me with the following statement:

    Proof that there are (one variable) polynomials of any natural degree with rational coefficients, which are irreducible over the field of rational numbers.

    Thanks.
    In other words, show that, given any n> 0, there exist a polynomial of degree n which has no rational root.

    I would suggest looking at x^n- p= 0 where p is a prime number.
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    In other words, show that, given any n> 0, there exist a polynomial of degree n which has no rational root.
    "no root" implies "irreducible" is true only if n<=3. It is a really common mistake. However your example still works due to Eisenstein's criterion.
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