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Math Help - Field of char p>0 & splitting field

  1. #1
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    Field of char p>0 & splitting field

    I'd appreciate any hints on how to prove the following:


    (a) Let F be a field of characteristic p > 0. Show that f = t4+1 ∈ F[t] is not irreducible.

    (b) Let K be a splitting field of f over F. Determine which finite field F must contain so
    that K = F.


    THANK YOU!
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  2. #2
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    Quote Originally Posted by mcasolin View Post

    I'd appreciate any hints on how to prove the following:

    (a) Let F be a field of characteristic p > 0. Show that f(t) = t^4+1 \in F[t] is not irreducible.
    if there exists a \in F such that a^2=-1, then we have (t^2-a)(t^2+a)=t^4+1. if there exists a \in F such that a^2=2, then (t^2 - at + 1)(t^2+at+1)=t^4 + 1.

    otherwise, there exists a \in F such that a^2=-2 and then t^4+1=(t^2-at-1)(t^2+at-1).


    (b) Let K be a splitting field of f over F. Determine which finite field F must contain so that K = F.
    read what you wrote again and see if you really understand it!!
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  3. #3
    Moo
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    (b) Let K be a splitting field of f over F. Determine which finite field F must contain so that K = F.
    read what you wrote again and see if you really understand it!!
    But it's understandable, isn't it ?

    Which finite field must F contain, so that K=F ?


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