GALOIS THEORY EXAM SCRIPT SOLVED
QUESTIONS ACCURATE ANSWER
COLLECTION
●● What is the degree [L : K] of a field extension?
Answer: The dimension of L as a K-vector space: [L : K] = dim_K(L). If
this is finite, L : K is called a finite extension.
●● What does it mean to adjoin an element α to K?
Answer: K(α) is the smallest subfield of L containing K and α. It equals
the set of all rational expressions in α with coefficients in K.
●● What is the rational function field K(t)?
Answer: The field of all rational functions f(t)/g(t) where f, g ∈ K[x] and
g ≠ 0. It is a transcendental extension of K.
●● What is a simple extension?
Answer: An extension of the form K(α) generated by a single element α.
●● What does it mean for α ∈ L to be algebraic over K?
Answer: There exists a nonzero polynomial f(t) ∈ K[t] such that f(α) = 0.
, ●● What does it mean for α ∈ L to be transcendental over K?
Answer: No nonzero polynomial f(t) ∈ K[t] satisfies f(α) = 0.
●● What is the minimal polynomial of α over K?
Answer: The unique monic irreducible polynomial m(t) ∈ K[t] of least
degree such that m(α) = 0. Its degree equals [K(α) : K].
●● What is an algebraic extension L : K?
Answer: Every element α ∈ L is algebraic over K.
●● What is a finitely generated extension L : K?
Answer: L = K(α₁,...,αₙ) for finitely many elements αᵢ ∈ L.
●● State the relation between finite, algebraic, and finitely generated
extensions.
Answer: Finite ⟹ algebraic. Finite ⟺ finitely generated by algebraic
elements. Algebraic does NOT imply finite in general.
●● What is a splitting field of f ∈ K[x] over K?
Answer: The smallest extension L of K over which f factors into linear
factors. Equivalently, L = K(α₁,...,αₙ) where α₁,...,αₙ are all roots of f.
●● What does it mean for L : K to be a normal extension?
QUESTIONS ACCURATE ANSWER
COLLECTION
●● What is the degree [L : K] of a field extension?
Answer: The dimension of L as a K-vector space: [L : K] = dim_K(L). If
this is finite, L : K is called a finite extension.
●● What does it mean to adjoin an element α to K?
Answer: K(α) is the smallest subfield of L containing K and α. It equals
the set of all rational expressions in α with coefficients in K.
●● What is the rational function field K(t)?
Answer: The field of all rational functions f(t)/g(t) where f, g ∈ K[x] and
g ≠ 0. It is a transcendental extension of K.
●● What is a simple extension?
Answer: An extension of the form K(α) generated by a single element α.
●● What does it mean for α ∈ L to be algebraic over K?
Answer: There exists a nonzero polynomial f(t) ∈ K[t] such that f(α) = 0.
, ●● What does it mean for α ∈ L to be transcendental over K?
Answer: No nonzero polynomial f(t) ∈ K[t] satisfies f(α) = 0.
●● What is the minimal polynomial of α over K?
Answer: The unique monic irreducible polynomial m(t) ∈ K[t] of least
degree such that m(α) = 0. Its degree equals [K(α) : K].
●● What is an algebraic extension L : K?
Answer: Every element α ∈ L is algebraic over K.
●● What is a finitely generated extension L : K?
Answer: L = K(α₁,...,αₙ) for finitely many elements αᵢ ∈ L.
●● State the relation between finite, algebraic, and finitely generated
extensions.
Answer: Finite ⟹ algebraic. Finite ⟺ finitely generated by algebraic
elements. Algebraic does NOT imply finite in general.
●● What is a splitting field of f ∈ K[x] over K?
Answer: The smallest extension L of K over which f factors into linear
factors. Equivalently, L = K(α₁,...,αₙ) where α₁,...,αₙ are all roots of f.
●● What does it mean for L : K to be a normal extension?
