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Math Help - Factoring given a root

  1. #1
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    Factoring given a root

    Suppose that \beta is a zero of f(x)=x^4+x+1 in some field extension E of \mathbb{Z}_2. Write f(x) as a product of linear factors in E[x].

    Since \beta is a zero, x-\beta is a factor of f(x). My plan was to simply divide x-\beta into f(x) to obtain a cubic and then find a root of the cubic, etc. until I obtained all the linear factors. Is there a reason that x-\beta shouldn't divide f(x)? I don't see one but it is not dividing it when I try. Does the fact that \beta lies in E and not \mathbb{Z}_2 affect this?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Factoring given a root

    Quote Originally Posted by AlexP View Post
    Suppose that \beta is a zero of f(x)=x^4+x+1 in some field extension E of \mathbb{Z}_2. Write f(x) as a product of linear factors in E[x].
    \beta satisfies \beta^4+\beta+1=0 in E . But (\beta^2)^4+\beta^2+1=(\beta^4+\beta+1)^2=0 so, \beta^2 is another root of f(x) in E . Now, prove that 1+\beta and 1+\beta^2 are also roots of f(x) in E . That is, f(x)=(x-\beta)(x-\beta^2)(x-1-\beta)(x-1-\beta^2) .
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  3. #3
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    Re: Factoring given a root

    I got it. Since we're in characteristic 2 we have f(a)=f(a+1) for arbitrary a (I did work out the details).

    Question about something I'm not clear on though... My book (Contemporary Abstract Algebra, Gallian, 5th ed.) says "A field E is an extension field of a field F if F \subseteq E and the operations of F are those of E restricted to F." That wording bothers me... Is that saying that E inherits the operations of F (and thus has the same characteristic?)? That's not what it seems to be saying to me, but it's what seems to make sense.
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    MHF Contributor Drexel28's Avatar
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    Re: Factoring given a root

    Quote Originally Posted by AlexP View Post
    I got it. Since we're in characteristic 2 we have f(a)=f(a+1) for arbitrary a (I did work out the details).

    Question about something I'm not clear on though... My book (Contemporary Abstract Algebra, Gallian, 5th ed.) says "A field E is an extension field of a field F if F \subseteq E and the operations of F are those of E restricted to F." That wording bothers me... Is that saying that E inherits the operations of F (and thus has the same characteristic?)? That's not what it seems to be saying to me, but it's what seems to make sense.
    More generally, F is an extension of E if there is an embedding \sigma:E\hookrightarrow F. In particular, since \sigma is an injective group morphism we know that \text{char}(F)=|1_F|=|\sigma(1_E)|=|1_E|=\text{cha  r}(E).
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  5. #5
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    Re: Factoring given a root

    Got it. Thanks to both of you.
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