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Thread: 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

    $\displaystyle 2x^2+4x+2$
    $\displaystyle 4x^2 +12x+8$

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


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




    These are both false, but Everything i try works for them...
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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

    $\displaystyle 2x^2+4x+2$
    $\displaystyle 4x^2 +12x+8$

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


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


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


    $\displaystyle (x^2+1)^5$ has no roots over $\displaystyle \mathbb{R}$, 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 $\displaystyle > 2x+2$

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

    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 $\displaystyle f(x) \in k[x]$ has a factorization

    $\displaystyle f(x)=ap_1(x)....p_t(x)$

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

    Let $\displaystyle k$ be a field and let $\displaystyle f(x) \in k[x]$ be a quadratic or cubic polynomial. Then $\displaystyle f(x)$ is irreducible in k[x] if and only if $\displaystyle f(x)$ does not have a root in $\displaystyle k$
    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...

    $\displaystyle x^4+2x+1 = (x^2+1)^2$

    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...

    $\displaystyle x^4+2x+1 = (x^2+1)^2$

    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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