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Finite fields

Source: RMO 2003, District Round

June 3, 2006
algebrapolynomialquadraticssuperior algebrasuperior algebra unsolved

Problem Statement

Let K\displaystyle \mathcal K be a finite field such that the polynomial X2−5\displaystyle X^2-5 is irreducible over K\displaystyle \mathcal K. Prove that: (a) 1+1≠01+1 \neq 0; (b) for all a∈K\displaystyle a \in \mathcal K, the polynomial X5+a\displaystyle X^5+a is reducible over K\displaystyle \mathcal K. Marian Andronache [Edit 1∘1^\circ] I wanted to post it in "Superior Algebra - Groups, Fields, Rings, Ideals", but I accidentally put it here :blush: Can any mod move it? I'd be very grateful. [Edit 2∘2^\circ] OK, thanks.
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