defs 6

Question 1

(Kronecker’s Theorem)

Let K be a field and let f ∈ K[X] be …….
Define …….. Then:

(i) L is a field and ……….
………
induces an ………..

(ii) α = ……. is a ……….;

(iii) L is a ……………,
with …………,
so …………….

Answer 1

Let K be a field and let f ∈ K[X] be irreducible of degree n.
Define L = K[X]/ < f >.Then:

(i) L is a field and the canonical homomorphism
π : K[X] → K[X]/
induces an embedding θ : K → L;

(ii) α = X+ < f > is a root of f in L;
(iii) L is a vector space over K of dimension n,

with {1, α, α^2, . . . , α^n−1} being a basis of L over K,

so every element of L has a unique representation of the form a_0 + a_1α + · · · + a_n−1α^n−1 with a_0, a_1, . . . , a_n−1 ∈ K.