Galois Theory

SOLUTION MANUAL For Galois Theory 5th Edition by
Ian Stewart

Explain The Following Terms?

Extension - ANSWER: If E is a field and F is a subfield of E. We use the notation E:F

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 - ANSWER: V a vector space over F. {A_i}_i∈I where each
vector in V can be expressed as a linear combinations of a finite number of vectors
from {A_i}_i∈I

Finite dimensional - ANSWER: V if it has a finite spanning set

, Finite Extension - ANSWER: An extension E/F is called finite if E is a finite dimensional
vector space over F

Degree of an Extension - ANSWER: The dimension of E over F. We use the notation (E
: F) = dim_F(E)

Tower Law - ANSWER: Let F, B, E be fields with F ⊂ B ⊂ E such that B/F and E/B are
finite extensions. Then E/F is a finite extension and
(E : F) = (E : B)(B : F)

Monic polynomial - ANSWER: A polynomial f with deg f = n and a_n = 1

Reducible - ANSWER: A non-constant polynomial f ∈ F[x] is this over F if f = gh for
some non-constant polynomials g, h ∈ F[x]

Uniquely determined - ANSWER: When exactly one thing satisfies some given
conditions

Minimum Polynomial - ANSWER: E:F, α ∈ E, α algebraic over F. f is this if
i) f is uniquely determined by α
ii) f is irreducible
iii) if g ∈ F[x] and g(α) = 0, then f|g

Algebraic element - ANSWER: α ∈ E is this, if α is aroot of a polynomial f ∈ F[x].
Otherwise, α is said to be transcendental
over F

Splitting Field - ANSWER: An extension E of F is this for p over F if p splits over E, but
p does not split over any intermediate field properly contained in E

Splits - ANSWER: Let p ∈ F[x] with deg p ≥ 1 and let E be an extension of F. We say
that p does this to mean that p can be factored into linear factors over E

Character - ANSWER: This of G in F is a function σ : G → F such that σ(xy) = σ(x)σ(y)
for all x, y ∈ G and σ(x) != 0 for all x ∈ G. In other words, σ is a (group)
homomorphism of G into the multiplicative group of the field F

Monomorphism - ANSWER: From a field E into a field E' is an injective map σ : E → E'
such that for all α, β ∈ E
σ(α + β) = σ(α) + σ(β)
σ(αβ) = σ(α)σ(β)

Fixed Point - ANSWER: If Σ = {ϕ_1, . . . ϕ_n} is a set of maps from E to itself (ϕ_i : E →
E), we say that a ∈ E is this for Σ if ϕ_i(a) = a for i = 1, . . . , n. It will be convenient to
generalise this notion to systems of maps between different sets: If Σ = {ϕ_1, . . .
ϕ_n} is a set of maps from E to E', we say that a ∈ E is a fixed point for Σ if