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Tom Leinster

Galois Theory

Galois Theory

This solution manual corresponds to the lecture notes accessible on Tom Leinster's website via the following link: \href{https://www.maths.ed.ac.uk/~tl/gt/gt.pdf}{https://www.maths.ed.ac.uk/~tl/gt/gt.pdf}

Chapter 1. Overview of Galois Theory

Exercise 1.1.10 (Conjugacy of Complex K-Tuples)

Exercise 1.1.3 (Conjugacy)

Exercise 1.1.6 (Conjugacy of Rational)

Exercise 1.2.2 (Galois Group as a Subgroup)

Chapter 10. Finite Fields

Exercise 10.1.4 (Frobenius automorphism on F3)

Exercise 10.1.7 (Verify field $\mathbb{F}_2(\alpha)$)

Exercise 10.2.3 (Finite subgroup of multiplicative group)

Exercise 10.2.6 (Extension over finite field is simple)

Exercise 10.3.10 (Diagrams)

Exercise 10.3.4 (Fixed field for Galois group of finite field)

Exercise 10.3.5 (Unique subgroup of cyclic group)

Chapter 2. Group actions, rings and fields

Exercise 2.1.10 (Isometry Cube)

Exercise 2.1.3 (Group Action Determines Bijection)

Exercise 2.2.14 (Ideal is equal to principal ideal)

Exercise 2.2.15 (Integral Domain Equivalencies)

Exercise 2.2.6 (Subring of ring that is an Ideal is same ring )

Exercise 2.2.8 (Trivial Ring)

Exercise 2.3.1 (Easy fields)

Exercise 2.3.13 (Homomorphism preserves characteristic)

Exercise 2.3.15 (Prime subfield)

Exercise 2.3.25 (Irreducible elements field)

Exercise 2.3.5 (Homomorphic inverse)

Chapter 3. Polynomials

Exercise 3.1.13 (pth root)

Exercise 3.1.4 (Polynomials and functions)

Exercise 3.1.8 (Substitution of polynomials)

Exercise 3.2.4 (Not principal ideals)

Exercise 3.3.13 (Eisenstein’s criterion proof)

Exercise 3.3.15 (Eisenstein's criterion)

Exercise 3.3.5 (Quadratic reducibility)

Chapter 4. Field extensions

Exercise 4.1.11 (Adjoined Expansion)

Exercise 4.1.3 (Examples)

Exercise 4.1.5 (Generated subfield)

Exercise 4.1.7 (Generated Subfield Examples)

Exercise 4.2.2 (All of K is Algebraic over K)

Exercise 4.2.9 (Minimal polynomial)

Exercise 4.3.15 (Simple extension)

Exercise 4.3.18 (Elements of field extension)

Exercise 4.3.5 (Minimal polynomial of homomorphism)

Exercise 4.3.9 (Uniqueness of homomorphism over K)

Chapter 5. Degree

Exercise 5.1.13 (Degree Inequality)

Exercise 5.1.16 (Polynomial of transcendental)

Exercise 5.1.20 (Prime degree is simple)

Exercise 5.1.23 (Coprime relation)

Exercise 5.1.9 (Tables)

Exercise 5.2.5 (K-linear subspace)

Exercise 5.2.8 (Algebraic elements form subfield)

Exercise 5.3.7 (Subfield lemma)

Chapter 6. Splitting fields

Exercise 6.1.5 (Properties of induced homomorphism)

Exercise 6.2.12 (Number of extensions)

Exercise 6.2.7 (Splitting field equivalence)

Exercise 6.2.9 (Splitting field example)

Exercise 6.3.11 (Galois group property)

Exercise 6.3.2 (Galois group defines a group)

Exercise 6.3.4 (Galois group third root of unity)

Chapter 7. Preparation for the fundamental theorem

Exercise 7.1.12 (Transitivity of the action over the roots)

Exercise 7.1.17 (Geometric description of Galois group)

Exercise 7.1.4 (Normal extensions lemma)

Exercise 7.2.1 (Irreducible over rationals)

Exercise 7.2.15 (Algebraic extensions)

Exercise 7.2.8 (Formal derivative)

Exercise 7.3.2 (Automorphism over prime subfield)

Exercise 7.3.5 (Fixed fields)

Chapter 8. The fundamental theorem of Galois theory

Exercise 8.1.4 (Galois correspondence)

Exercise 8.1.5 (Galois correspondence diagram)

Exercise 8.1.8 (Galois and Fix not mutually inverse)

Exercise 8.2.6 (Normality)

Exercise 8.3.3 (Subgoups of $D_4$)

Exercise 8.3.4 (Fixed field)

Exercise 8.3.5 (Fundamental theorem isomorphism)

Chapter 9. Radicals

Exercise 9.1.10 (Eigenvectors of automorphisms)

Exercise 9.1.11 (Solvability of Galois group)

Exercise 9.1.3 ((Radical) subfield)

Exercise 9.1.7 (Abelian Galois group)

Exercise 9.2.2 (Solvability of extensions)

Exercise 9.2.5 (Extension solvable iff Galois group is solvable)

Exercise 9.3.4 (Order of permutation cycle)

Exercise 9.3.6 (Polynomial unsolvable by radicals)