1. Let E/F be a field extension and let a∈E be algebraic over F. Prove the following a) Let f∈F[X] be irreducible for wh
Posted: Fri Apr 29, 2022 11:28 am
1. Let E/F be a field extension and let a∈E be algebraic over F.
Prove the following
a) Let f∈F[X] be irreducible for which degree of f has degree
greater than or equal to 2 and gcd(deg(f),[F(a):F])=1. Show that f
has no roots in F(a)
b) Show F[a]=F(a) for each f∈F[X] and f(a) not equal to 0 we get
f(a)-1 (basically showing how to compute
f(a)-1)
Prove the following
a) Let f∈F[X] be irreducible for which degree of f has degree
greater than or equal to 2 and gcd(deg(f),[F(a):F])=1. Show that f
has no roots in F(a)
b) Show F[a]=F(a) for each f∈F[X] and f(a) not equal to 0 we get
f(a)-1 (basically showing how to compute
f(a)-1)