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MTH416Sciences3 Unitsadvanced

Algebraic Number Theory

This course, Algebraic Number Theory, delves into abstract mathematics, emphasizing its practical applications. It explores algebraic numbers, factorization, and irreducibility using Eisenstein's Theorem. Students will learn about ideals, prime ideals, class groups, and class numbers. The course also covers Fermat's Last Theorem, Dirichlet's Theorem, and Minkowski's Theorem, fostering abstract thinking and preparing students for advanced studies in number theory.

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150h

Study Time

13

Weeks

12h

Per Week

advanced

Math Level

Course Keywords

Algebraic NumbersEisenstein TheoremIdealsFermat's Last TheoremDirichlet's Theorem

Course Overview

Everything you need to know about this course

Course Difficulty

Advanced Level

For experienced practitioners

90%

advanced

∑

Math Level

Advanced Math

📖

Learning Type

Theoretical Focus

Course Topics

Key areas covered in this course

1

Rings and Fields

2

Algebraic Numbers and Elements

3

Quadratic and Cyclotomic Fields

4

Factorization of Polynomials

5

Ideals and Quotient Rings

6

Fermat's Last Theorem

7

Dirichlet's Theorem

8

Minkowski's Theorem

Total Topics8 topics

Requirements

Knowledge and skills recommended for success

Abstract Algebra I

Abstract Algebra II

💡 Don't have all requirements? Don't worry! Many students successfully complete this course with basic preparation and dedication.

Assessment Methods

How your progress will be evaluated (3 methods)

assignments

Comprehensive evaluation of course material understanding

Written Assessment

tutor-marked assessments

Comprehensive evaluation of course material understanding

Written Assessment

final examination

Comprehensive evaluation of course material understanding

Written Assessment

Career Opportunities

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Theoretical Mathematician

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Cryptography Researcher

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Algorithm Developer

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Industry Applications

Real-world sectors where you can apply your knowledge

CryptographyData SecurityTelecommunicationsFinancial ModelingResearch and Development

Study Schedule Beta

A structured 13-week journey through the course content

Week

1

Module 1: Algebraic Numbers

5h

Unit 1: Ring

3 study hours

•Understand the definition of a ring and its properties.
•Study examples of commutative rings with and without identity.
•Solve exercises to verify ring properties.

Unit 1: Ring

2 study hours

•Learn about zero divisors and proper zero divisors.
•Determine if given rings have proper zero divisors.
•Understand the concept of an integral domain.

Week

2

Module 1: Algebraic Numbers

4h

Unit 2: Field

4 study hours

•Understand the definition of a field and its properties.
•Verify if given sets form a field.
•Solve exercises related to multiplicative inverses in fields.

Week

3

Module 1: Algebraic Numbers

5h

Unit 3: Algebraic Numbers (extension field)

5 study hours

•Define extension fields and algebraic elements.
•Determine if a number is algebraic over a given field.
•Work through examples to identify algebraic and transcendental elements.

Week

4

Module 2: Quadratic and Cyclotomic Fields

4h

Unit 1: Quadratic Field

4 study hours

•Define quadratic fields and their elements.
•Perform addition and multiplication operations in quadratic fields.
•Solve numerical examples to practice these operations.

Week

5

Module 2: Quadratic and Cyclotomic Fields

3h

Unit 1: Quadratic Field

3 study hours

•Understand the concept of a square-free integer.
•Learn about quadratic integers and how to identify them.
•Work through exercises to compute norms in quadratic fields.

Week

6

Module 2: Quadratic and Cyclotomic Fields

5h

Unit 2: Cyclotomic Field

5 study hours

•Define ideals, principal ideals, and prime ideals.
•Understand the concept of nth root of unity.
•Study examples of cyclotomic fields.

Week

7

Module 3: Factorization into irreducible and ideals

4h

Unit 1: Factorization of Polynomials over a Field

4 study hours

•Review the definition of a polynomial and its coefficients.
•Understand the Division Algorithm for polynomials.
•Work through computational examples using synthetic division.

Week

8

Module 3: Factorization into irreducible and ideals

5h

Unit 2: Factorizing into irreducible

5 study hours

•Define irreducible polynomials and their properties.
•State and apply Eisenstein's Theorem to determine irreducibility.
•Work through examples to apply the theorem.

Week

9

Module 3: Factorization into irreducible and ideals

4h

Unit 1: Ideals

4 study hours

•Understand the definition of an ideal in a ring.
•Learn about two-sided, left, and right ideals.
•Work through examples to identify ideals in given rings.

Week

10

Module 3: Factorization into irreducible and ideals

5h

Unit 2: Class Group and Class Number

5 study hours

•Define class groups and class numbers.
•Understand the concept of a discriminant.
•Study the formula for class numbers of quadratic orders.

Week

11

Module 4: Fermat's Last Theorem, Dirichilet Theorem and Minkowski's

4h

Unit 1: Fermat's Last Theorem

4 study hours

•Understand Fermat's Last Theorem and its history.
•Review the proof for the case n=4.
•Study the implications of the theorem.

Week

12

Module 4: Fermat's Last Theorem, Dirichilet Theorem and Minkowski's

5h

Unit 2: Dirichlet's and Minkowski's Theorems

5 study hours

•State Dirichlet's Theorem on Diophantine approximation.
•Understand Minkowski's Theorem.
•Study examples illustrating Dirichlet's Theorem.

Week

13

Module 4: Fermat's Last Theorem, Dirichilet Theorem and Minkowski's

6h

Unit 2: Dirichlet's and Minkowski's Theorems

6 study hours

•Review all modules and units.
•Work on practice problems and exercises.
•Prepare for assignments and tutor-marked assignments.

This study schedule is in beta and may not be accurate. Please use it as a guide and consult the course outline for the most accurate information.

Course PDF Material

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Study Tips & Exam Preparation

Expert tips to help you succeed in this course

1

Review definitions and properties of rings, fields, and integral domains from Unit 1.

2

Practice applying Eisenstein's Theorem to various polynomials (Units 7-8).

3

Work through examples of operations in quadratic and cyclotomic fields (Units 4-6).

4

Focus on understanding the key concepts and theorems related to ideals (Unit 9).

5

Study the historical context and implications of Fermat's Last Theorem (Unit 11).

6

Understand the statements and applications of Dirichlet's and Minkowski's Theorems (Unit 12).

7

Create concept maps linking algebraic structures and their properties.

8

Solve all exercises and tutor-marked assignments to reinforce understanding.

9

Allocate study time evenly across all modules, with extra focus on challenging units.

10

Form study groups to discuss concepts and solve problems collaboratively.

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