GALOIS THEORY CERTIFICATION
EVALUATION VERIFIED QUESTIONS WITH
COMPLETE SOLUTIONS
●● Field
Answer: A set F with binary operations + and x such that
i) (F, +) is an abelian group with identity element 0
ii) (F\{0}, x) is an abelian group with identity element 1
iii) x distributes over +
All elements are invertible
●● Subfield
Answer: A set within a field which itself is a field under the binary
operations of the original field
●● F-automorphism of E
Answer: Automorphisms Φ of E such that Φ(a) = a, ∀ a ∈ F
●● Characteristic
Answer: The smallest n ∈ ℕ such that
n.1 = 1 + 1 + ... + 1 = 0
char F = n
, ●● Prime subfield
Answer: The smallest subfield of F
●● Vector Space
Answer: An additively written abelian group V with scalar
multiplication F x V -> V (a ∈ F, A ∈ V, (a, A) ↔ aA, aA ∈ V) such that
∀ a, b ∈ F; A, B ∈ V
i) a(A + B) = aA + aB
ii) (a + b)A = aA +bA
iii) a(bA) = (ab)A
iV) 1 A = A
●● Linearly dependent
Answer: V vector space over a field F. {A_1, ..., A_n} ⊆ V is this if ∃
a_1, ..., a_n ∈ F, not all zero such that
a_1 A_1 + a_2 A_2 + ... + a_n A_n = zero vector
●● Linearly independent
Answer: The set containing two or more vectors which are not multiples
is..
●● Generating/spanning set
EVALUATION VERIFIED QUESTIONS WITH
COMPLETE SOLUTIONS
●● Field
Answer: A set F with binary operations + and x such that
i) (F, +) is an abelian group with identity element 0
ii) (F\{0}, x) is an abelian group with identity element 1
iii) x distributes over +
All elements are invertible
●● Subfield
Answer: A set within a field which itself is a field under the binary
operations of the original field
●● F-automorphism of E
Answer: Automorphisms Φ of E such that Φ(a) = a, ∀ a ∈ F
●● Characteristic
Answer: The smallest n ∈ ℕ such that
n.1 = 1 + 1 + ... + 1 = 0
char F = n
, ●● Prime subfield
Answer: The smallest subfield of F
●● Vector Space
Answer: An additively written abelian group V with scalar
multiplication F x V -> V (a ∈ F, A ∈ V, (a, A) ↔ aA, aA ∈ V) such that
∀ a, b ∈ F; A, B ∈ V
i) a(A + B) = aA + aB
ii) (a + b)A = aA +bA
iii) a(bA) = (ab)A
iV) 1 A = A
●● Linearly dependent
Answer: V vector space over a field F. {A_1, ..., A_n} ⊆ V is this if ∃
a_1, ..., a_n ∈ F, not all zero such that
a_1 A_1 + a_2 A_2 + ... + a_n A_n = zero vector
●● Linearly independent
Answer: The set containing two or more vectors which are not multiples
is..
●● Generating/spanning set
