# Field of char p>0 & splitting field

• Apr 21st 2009, 11:20 AM
mcasolin
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!
• Apr 22nd 2009, 12:00 AM
NonCommAlg
Quote:

Originally Posted by mcasolin

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

(a) Let F be a field of characteristic p > 0. Show that $\displaystyle f(t) = t^4+1 \in F[t]$ is not irreducible.

if there exists $\displaystyle a \in F$ such that $\displaystyle a^2=-1,$ then we have $\displaystyle (t^2-a)(t^2+a)=t^4+1.$ if there exists $\displaystyle a \in F$ such that $\displaystyle a^2=2,$ then $\displaystyle (t^2 - at + 1)(t^2+at+1)=t^4 + 1.$

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

Quote:

(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!!
• Apr 22nd 2009, 12:20 AM
Moo
Quote:

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

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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