Let $\mathbb { F } _ { q }$ be the finite field with $q$ elements and $P \in \mathbb { F } _ { q } [ x ]$ be a monic irreducible polynomial of even degree $2 d$. Then show that $P$, when considered as a polynomial in $\mathbb { F } _ { q ^ { 2 } } [ x ]$, decomposes into a product $P = Q _ { 1 } Q _ { 2 }$ of irreducible polynomials $Q _ { i }$ in $\mathbb { F } _ { q ^ { 2 } } [ x ]$ with $\operatorname { deg } \left( Q _ { i } \right) = d$.