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  1. MHB Master
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    #1
    Hey!!

    In my notes there is the following:

    Let $F$ be a field. The irresducible $f\in F[x]$ is separable, if all the roots are different.
    A non-constant polynomial $f\in K[x]$ is separable, if all the irreducible factors are separable.

    Example:
    $f(x)=(x^2-2)^2(x^2+3)\in \mathbb{Q}[x]$.
    The irreducible factors are:
    • $x^2-2$ $\rightarrow$ Roots : $\pm 2$ different
    • $x^2+3$ $\rightarrow$ Roots : $\pm i\sqrt{3}$ different

    So, $f$ is separable.


    I haven't understood why $f$ is separable. Since we have the factor $(x^2-2)^2$ aren't the roots $\pm 2$ of multiplicity $2$ ?

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    #2
    Quote Originally Posted by mathmari View Post
    Hey!!

    In my notes there is the following:

    Let $F$ be a field. The irresducible $f\in F[x]$ is separable, if all the roots are different.
    A non-constant polynomial $f\in K[x]$ is separable, if all the irreducible factors are separable.

    Example:
    $f(x)=(x^2-2)^2(x^2+3)\in \mathbb{Q}[x]$.
    The irreducible factors are:
    • $x^2-2$ $\rightarrow$ Roots : $\pm 2$ different
    • $x^2+3$ $\rightarrow$ Roots : $\pm i\sqrt{3}$ different

    So, $f$ is separable.


    I haven't understood why $f$ is separable. Since we have the factor $(x^2-2)^2$ aren't the roots $\pm 2$ of multiplicity $2$ ?
    Hey mathmari!!

    Aren't the roots $\pm \sqrt 2$?

    And isn't $f(x)$ reducible? That would mean that the first statement doesn't apply.
    I think we should apply the second statement first.

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    #3 Thread Author
    Quote Originally Posted by I like Serena View Post
    Aren't the roots $\pm \sqrt 2$?
    Ah yes...


    Quote Originally Posted by I like Serena View Post
    And isn't $f(x)$ reducible? That would mean that the first statement doesn't apply.
    I think we should apply the second statement first.
    Yes, we apply the second statement first. The irreducible factors $x^2-2$ and $x^2+3$, right?

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    #4 Thread Author
    So, an inseparable polynomial is for example $f(x)=(x^2-4)^2(x^3-8)$, where the irreducible factors are $x^2-4$ and $x^3-8$, and the roots are $\pm 2$ and $2$ ?

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    #5
    Quote Originally Posted by mathmari View Post
    Yes, we apply the second statement first. The irreducible factors $x^2-2$ and $x^2+3$, right?
    Yes. That's what your example states as well.

    Quote Originally Posted by mathmari View Post
    So, an inseparable polynomial is for example $f(x)=(x^2-4)^2(x^3-8)$, where the irreducible factors are $x^2-4$ and $x^3-8$, and the roots are $\pm 2$ and $2$?
    Can't we reduce $x^3-8$ to $(x-2)(x^2+2x+4)$?

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    #6 Thread Author
    Quote Originally Posted by I like Serena View Post
    Can't we reduce $x^3-8$ to $(x-2)(x^2+2x+4)$?
    Ah, so it is again separable, right?

    Could you give me an example of an inseparable polynomial?

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    #7
    Quote Originally Posted by mathmari View Post
    Could you give me an example of an inseparable polynomial?
    How about $x^3-2$ in $F_3[x]$?

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    #8 Thread Author
    Quote Originally Posted by I like Serena View Post
    How about $x^3-2$ in $F_3[x]$?
    I saw now in my notes the following proposition:
    If $F$ is finite, then each non-constant $f\in F[x]$ is separable.

    We have that $F_3$ is finite, right? Therefore, $x^3-2$ is separable, isn't it?

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    #9
    Quote Originally Posted by mathmari View Post
    I saw now in my notes the following proposition:
    If $F$ is finite, then each non-constant $f\in F[x]$ is separable.

    We have that $F_3$ is finite, right? Therefore, $x^3-2$ is separable, isn't it?
    Well... $x^3-2$ is irreducible, but we also have $x^3-2=(x+1)^3$ in $F_3[x]$, so $2$ is a triple root.
    If that doesn't mean it's inseparable, I wouldn't know what is.

    Do your notes give an example of an inseparable polynomial?
    Or do you have a proof for that proposition?

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    #10 Thread Author
    Quote Originally Posted by I like Serena View Post
    Well... $x^3-2$ is irreducible, but we also have $x^3-2=(x+1)^3$ in $F_3[x]$, so $2$ is a triple root.
    If that doesn't mean it's inseparable, I wouldn't know what is.

    Do your notes give an example of an inseparable polynomial?
    Or do you have a proof for that proposition?
    In this the chapter 3.4 (p.11) is about separability. This proposition is the 3.4.5 and the only example of an inseparable polynomial that I found is the one that is at 3.4.8.


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