Algebraic Number Theory
- Sciences
- 400 level
- 3 credit units
- 88 pages
- 11 units
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.
About this course
- Difficulty
- Advanced
- Study hours
- 150 hours
- Maths
- Advanced
- Content
- Theoretical, problem solving
- Practical work
- No
- Abstract Algebra I
- Abstract Algebra II
- Assignments
- Tutor marked assessments
- Final examination
What you'll read
The real module and unit structure of MTH416, taken from the course material NOUN publishes.
One paragraph, so you can see how it reads
MTH416 · Unit 1. Ring
Thirdly, even those who prefer analytic number theory will agree that the full generality and power of the analytic approach reveals itself only in the context of number fields and simple algebras, not in investigations involving the rational number alone.
What you should be able to do
- Define and apply the properties of rings, fields, and integral domains.
- Identify algebraic numbers and determine if a polynomial is irreducible.
- Perform operations in quadratic and cyclotomic fields.
- Understand and apply Eisenstein's criteria for irreducibility.
- Define and work with ideals, prime ideals, and quotient rings.
- Explain Fermat's Last Theorem and its historical context.
- State and apply Dirichlet's and Minkowski's Theorems.
What it prepares you for
- Cryptographer
- Data Security Analyst
- Theoretical Mathematician
- Cryptography Researcher
- Algorithm Developer
- Cryptography
- Data Security
- Telecommunications
- Financial Modeling
- Research and Development
Where it gets hard
The units students slow down on, and what makes each one heavy.
- Module 3: Factorization into irreducible and ideals
Unit 2: Factorizing into irreducible
Application of Eisenstein's Theorem requires careful attention to the conditions and prime number selection.
- Module 3: Factorization into irreducible and ideals
Unit 1: Ideals
Understanding the relationships between ideals, quotient rings, and factor rings requires a solid grasp of abstract algebra concepts.
- Module 4: Fermat's Last Theorem, Dirichilet Theorem and Minkowski's
Unit 1: Fermat's Last Theorem
The proof of Fermat's Last Theorem involves advanced concepts in number theory and requires a strong mathematical background.
A suggested way through it
13 weeks, about 59 hours in total. Yours will differ.
- Week 1Module 1: Algebraic Numbers
Unit 1: Ring · 3 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 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 2Module 1: Algebraic Numbers
Unit 2: Field · 4 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 3Module 1: Algebraic Numbers
Unit 3: Algebraic Numbers (extension field) · 5 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 4Module 2: Quadratic and Cyclotomic Fields
Unit 1: Quadratic Field · 4 hours
Define quadratic fields and their elements.. Perform addition and multiplication operations in quadratic fields.. Solve numerical examples to practice these operations..
- Week 5Module 2: Quadratic and Cyclotomic Fields
Unit 1: Quadratic Field · 3 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 6Module 2: Quadratic and Cyclotomic Fields
Unit 2: Cyclotomic Field · 5 hours
Define ideals, principal ideals, and prime ideals.. Understand the concept of nth root of unity.. Study examples of cyclotomic fields..
- Week 7Module 3: Factorization into irreducible and ideals
Unit 1: Factorization of Polynomials over a Field · 4 hours
Review the definition of a polynomial and its coefficients.. Understand the Division Algorithm for polynomials.. Work through computational examples using synthetic division..
- Week 8Module 3: Factorization into irreducible and ideals
Unit 2: Factorizing into irreducible · 5 hours
Define irreducible polynomials and their properties.. State and apply Eisenstein's Theorem to determine irreducibility.. Work through examples to apply the theorem..
- Week 9Module 3: Factorization into irreducible and ideals
Unit 1: Ideals · 4 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 10Module 3: Factorization into irreducible and ideals
Unit 2: Class Group and Class Number · 5 hours
Define class groups and class numbers.. Understand the concept of a discriminant.. Study the formula for class numbers of quadratic orders..
- Week 11Module 4: Fermat's Last Theorem, Dirichilet Theorem and Minkowski's
Unit 1: Fermat's Last Theorem · 4 hours
Understand Fermat's Last Theorem and its history.. Review the proof for the case n=4.. Study the implications of the theorem..
- Week 12Module 4: Fermat's Last Theorem, Dirichilet Theorem and Minkowski's
Unit 2: Dirichlet's and Minkowski's Theorems · 5 hours
State Dirichlet's Theorem on Diophantine approximation.. Understand Minkowski's Theorem.. Study examples illustrating Dirichlet's Theorem..
- Week 13Module 4: Fermat's Last Theorem, Dirichilet Theorem and Minkowski's
Unit 2: Dirichlet's and Minkowski's Theorems · 6 hours
Review all modules and units.. Work on practice problems and exercises.. Prepare for assignments and tutor-marked assignments..
Preparing for the exam
- Review definitions and properties of rings, fields, and integral domains from Unit 1.
- Practice applying Eisenstein's Theorem to various polynomials (Units 7-8).
- Work through examples of operations in quadratic and cyclotomic fields (Units 4-6).
- Focus on understanding the key concepts and theorems related to ideals (Unit 9).
- Study the historical context and implications of Fermat's Last Theorem (Unit 11).
- Understand the statements and applications of Dirichlet's and Minkowski's Theorems (Unit 12).
- Create concept maps linking algebraic structures and their properties.
- Solve all exercises and tutor-marked assignments to reinforce understanding.
- Allocate study time evenly across all modules, with extra focus on challenging units.
- Form study groups to discuss concepts and solve problems collaboratively.
Questions students ask about this course
What is MTH416 about?
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.
How many units does MTH416 have?
MTH416, Algebraic Number Theory, has 11 units across 4 modules, over 88 pages of course material. You can read it one unit at a time.
How many credit units is MTH416?
MTH416 carries 3 credit units, at 400 level in Sciences.
Is MTH416 hard?
MTH416 is rated advanced level, with advanced mathematical content. It is mostly theoretical and problem solving work.
How long does MTH416 take to study?
About 150 hours of study, spread across its 11 units.
How is MTH416 assessed?
MTH416 is assessed by assignments, tutor marked assessments and final examination.
What do I need before starting MTH416?
Abstract Algebra I Abstract Algebra II
What can I do with MTH416?
Cryptographer, Data Security Analyst, Theoretical Mathematician, Cryptography Researcher and Algorithm Developer.