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Math Help - counter examples!

  1. #1
    ux0
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    counter examples!

    I just need a counter example for each of theses problems, they have been killing me for a few days now..


    A) The gcd of

    2x^2+4x+2
    4x^2 +12x+8

    in \mathbb{Q}[x] is 2x+2


    B) If k is a field in p(x) \in k[x] is a non constant polynomial having no roots in k, then p(x) is irreducible in k[x]




    These are both false, but Everything i try works for them...
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  2. #2
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    Quote Originally Posted by ux0 View Post
    I just need a counter example for each of theses problems, they have been killing me for a few days now..


    A) The gcd of

    2x^2+4x+2
    4x^2 +12x+8

    in \mathbb{Q}[x] is 2x+2


    2x^2+4x+2=2(x+1)^2\,,\,4x^2+12x+8=4(x+1)(x+2), so the gcd is x+1 , up to a constant, and thus your answer is correct, though perhaps not the prettiest one.


    B) If k is a field in p(x) \in k[x] is a non constant polynomial having no roots in k, then p(x) is irreducible in k[x]


    (x^2+1)^5 has no roots over \mathbb{R}, but it is terribly NOT irreducible...

    Tonio

    These are both false, but Everything i try works for them...
    .
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  3. #3
    ux0
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    THose solution don't work, each of those statements are Actually False, so I need to find an example that proves them wrong...

    So in the first part, i need to find the GCD that's > 2x+2

    And in the second part, i need to find a poly, that is in a Field, k. Which has no roots in k, but it is reducible in the k[x]

    that's the problem I'm having...

    I'm thinking in part two I need to use the proposition:

    If k is a field, then every non constant polynomial f(x) \in k[x] has a factorization

    f(x)=ap_1(x)....p_t(x)

    where a is a nonzero constant and the p_i(x) are monic irreducible polynomials.
    or this proposition

    Let k be a field and let f(x) \in k[x] be a quadratic or cubic polynomial. Then f(x) is irreducible in k[x] if and only if f(x) does not have a root in k
    It is clear by the second proposition my counter example to part two would be of degree greater than 3...
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  4. #4
    ux0
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    oops found part two...

    x^4+2x+1 = (x^2+1)^2

    No real roots, but is reducible..


    Still don't have part 1... i was thinking it was like you said

    2x+2 is just a linear combo of x+1 therefore x+1 is unique, and monic, where 2x+2 is not unique.
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    Quote Originally Posted by ux0 View Post
    oops found part two...

    x^4+2x+1 = (x^2+1)^2

    No real roots, but is reducible..


    Still don't have part 1... i was thinking it was like you said

    2x+2 is just a linear combo of x+1 therefore x+1 is unique, and monic, where 2x+2 is not unique.

    I don't know what you're doing: in my previous post I answered your questions. You may believe what you want but unless proven wrong I'm certain 99.999% about their being right.

    Tonio
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