Homework SourseShow that the irreducible polynomial over R has degree one o
Show that the irreducible polynomial over R has degree one or degree two.
Solution
algebraic extension C K = C(a) with [K/C] = deg Ca = d > 1.
Then K = C(a) = R(i,a) = R(b) for some bK by the primitive element theorem
But then the minimal polynomial f(X) R[X] of b over R would be irreducible over R and have degree degf(X)=2d>2, a contradiction to our hypothesis.
![Show that the irreducible polynomial over R has degree one or degree two.Solutionalgebraic extension C K = C(a) with [K/C] = deg Ca = d > 1. Then K = C(a) =](/WebImages/41/show-that-the-irreducible-polynomial-over-r-has-degree-one-o-1127965-1761601914-0.webp "Show that the irreducible polynomial over R has degree one or degree two.Solutionalgebraic extension C K = C(a) with [K/C] = deg Ca = d > 1. Then K = C(a) =")
