Lecture 3 : Algebraic Extensions II
Objectives
(1) Degree of a field extension and its multiplicative nature.
(2) A field extension of finite degree is algebraic.
(3) Transitivity of algebraic extensions.
(4) Compositum of two fields.
Key words and phrase

Lecture 4 : Ruler and Compass Constructions I
Objectives
(1) Describe standard ruler and compass constructions.
(2) The field of constructible numbers is closed under taking square roots
of positive reals.
(3) Characterization of constructible real number

Lecture 24 : Galois Groups of Quartic Polynomials
Objectives
(1) Galois group as a group of permutations.
(2) Irreducibility and transitivity.
(3) Galois groups of quartics.
Keywords and phrases : Transitive subgroups of S4 Galois groups of
quartics, irre

Lecture 12 : The Primitive Element Theorem
Objectives
(1) Factorization of polynomials over finite fields.
(2) The Primitive element theorem.
(3) Finite separable extensions have a primitive element.
Key words and phrases: Primitive element, finite separa

Lecture 22 : Solvability by Radicals
Objectives
(1) Radical extensions.
(2) Solvability by radicals and solvable Galois groups.
(3) A quintic polynomial which is not solvable by radicals.
Keywords and Phrases : Radical extensions, solvable Galois groups,

Lecture 15 : Galois group of a Galois Extension II
Objectives
(1) Artins Theorem about fixed field of a finite group of automorphisms.
(2) Behavior of Galois group under isomorphisms.
(3) Normal subgroups of the Galois groups and their fixed fields.
Keywo

Lecture 10 : Separable Extensions II
Objectives
(1) Roots of an irreducible polynomial have equal multiplicity.
(2) Separable finite algebraic extensions and separable degree.
(3) Transitivity of separable extensions
Key words and phrases: Separable degre

Lecture 20 : Cyclic Extensions and Solvable Groups
Objectives
(1) Cyclic extensions of degree p over fields with characteristics p.
(2) Solvable groups.
(3) Simplicity of Sn and An .
Keywords and phrases : Cyclic extensions, solvable groups, commutator, s

Lecture 18 : Cyclotomic Extensions II
Objectives
(1) Discriminant of p (x).
(2) Subfields of Q(p ).
(3) Kronecker-Weber Theorem for quadratic extensions of Q.
(4) Algorithm for construction of primitive elements of subfields of Q(p ).
(5) Subfields of Q(7

Lecture 8 : Algebraic Closure of a Field
Objectives
(1) Existence and isomorphisms of algebraic closures.
(2) Isomorphism of splitting fields of a polynomial.
Key words and phrases: algebraically closed field, algebraic closure, splitting field.
In the pr

Lecture 26 : Polynomials with Galois Group Sn .
Objectives
(1) Tates proof of Dedekinds theorem for computing Galois group
(2) Construction of polynomials with Galois group Sn .
Keywords and phrases: Polyomials with Galois group Sn Dedekinds
reduction mod

Lecture 11 : Finite Fields I
Objectives
(1) Existence and uniqueness of finite fields.
(2) Algebraic closure of a finite field.
(3) Finite subgroup of the multiplicative group of a field is cyclic.
(4) Gauss formula for the number of monic irreducible pol

Lecture 19 : Abelian and Cyclic Extensions
Objectives
(1) Infinitude of primes p 1 ( mod n ).
(2) Inverse Galois problem for finite abelian groups.
(3) Structure of some cyclic extensions.
Keywords and phrases : Primes of the form p 1 ( mod n ), abelian
e

Lecture 6 : Symmetric Polynomials I
Objectives
(1) Examples of symmetric polynomials.
(2) The fundamental theorem of symmetric polynomials.
(3) Newtons identities for power sum symmetric polynomials.
Key words and phrases: Symmetric polynomial, symmetriza

Lecture 16 : Applications and Illustrations of the FTGT
Objectives
(1) Fundamental theorem of algebra via FTGT.
(2) Gauss criterion for constructible regular polygons.
(3) Symmetric rational functions.
(4) Galois group of some binomials.
Keywords and phra

Lecture 1 : Overview
Objectives
(1) A historical sketch of main discoveries about formulas for roots of
polynomials.
(2) Problems of classical Greek geometry.
(3) Discussion about the main themes of the course.
Key words and phrases: Quadratic, cubic and

Lecture 9 : Separable Extensions I
Objectives
(1) Criterion for multiple roots in terms of derivatives
(2) Irreducible polynomials are separable over fields of characteristic
zero.
(3) Characterization of perfect fields of positive characteristic,
Key wor

Lecture 2 : Algebraic Extensions I
Objectives
(1) Main examples of fields to be studied.
(2) The minimal polynomial of an algebraic element.
(3) Simple field extensions and their degree.
Key words and phrases: Number field, function field, algebraic eleme

Lecture 17 : Cyclotomic Extensions I
Objectives
(1) Roots of unity in a field.
(2) Galois group of xn a over a field having nth roots of unity.
(3) Irreduciblilty of the cyclotomic polynomial n (x) over Q.
(4) A recursive formula for n (x).
Keywords and p

Lecture 25 : Norm, Trace and Hilberts Theorem 90
Objectives
(1) The norm and the trace function.
(2) Multiplicative form of Hilberts Theorem 90.
(3) Cyclic extensions of degree n.
(4) Additive version of Hilberts 90.
(5) Cyclic extensions of prime degree:

Lecture 14 : Galois group of a Galois Extension I
Objectives
(1) Galois extension and the Galois group of a Galois extension.
(2) Galois group of a finite extension of finite fields and quadratic extensions.
(3) Galois groups of biquadratic extension.
(4)

Examination Papers
MA 414
Quiz 1
Duration
Max. Marks 10
40 minutes Weightage 10 %
(1) Show that if a regular polygon of p sides, where p is a prime number,
is constructible by ruler and compass, then p is a Fermat prime. [2]
(2) Let F be a field and x be

Lecture 7 : Symmetric Polynomials II
Objectives
(1) Discriminant in terms of power-sum symmetric polynomials.
(2) Discriminant of a cubic.
(3) Existence of a splitting field of a polynomial.
(4) Fundamental theorem of algebra via symmetric polynomials.
Ke

Lecture 5 : Ruler and Compass Constructions II
Objectives
(1) Wantzels characterization of constructible regular p-polygons.
(2) Richmonds construction of a regular pentagon.
(3) Gauss criterion of constructible regular polygons.
Key words and phrases: Fe

Problem set 10 : Solvability by Radicals
(1) Show that the polynomials f (x) = x5 14x + 7, g(x) = x5 7x2 + 7
h(x) = x7 10x5 + 15x + 5 and `(x) = x5 6x + 3 are not solvable
by radicals over Q.
(2) Let f (x) Q[x] be an irreducible polynomial of prime degree

Problem set 9 : Cyclotomic Extensions
(1) Determine [Q(7 , 3 ) : Q(3 )].
(2) Determine a primitive element of a subfield K of E = Q(13 ) so that
[K : Q] = 3.
(3) Put = 7 . Determine the degrees of + 5 and + 5 + 8 over Q.
(4) Put = 11 and = + 3 + 4 + 5 + 9

Problem set 4 : Splitting Fields
(1) Let F be a field and let K be a splitting field of a polynomial f (x)
F [x]. Show that [K : F ] n!.
(2) Find degrees of splitting fields over Q of each of the following polynomials: (a) x3 2 (b) x4 1 (c) x4 + 1 (d) x6