Exam (elaborations)
Galois Theory
- Course
- Institution
- Book
Exam study book Galois' Theory of Algebraic Equations of Jean-Pierre Tignol - ISBN: 9789810245412 (Galois Theory)
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
More summaries for
-
Galois Theory
-
Galois Theory
-
Summary Galois' Theory of Algebraic Equations - Mathematics
Seller
Follow
kushboopatel6867
Reviews received
Content preview
SOLUTION MANUAL For Galois Theory 5th Edition byIan 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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller kushboopatel6867. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $17.99. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75632 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now