Please note! Course description is confirmed for two academic years, which means that in general, e.g. Learning outcomes, assessment methods and key content stays unchanged. However, via course syllabus, it is possible to specify or change the course execution in each realization of the course, such as how the contact sessions are organized, assessment methods weighted or materials used.

LEARNING OUTCOMES

Credits: 5

Schedule: 19.04.2021 - 28.05.2021

Teacher in charge (valid 01.08.2020-31.07.2022): Pekka Alestalo

Teacher in charge (applies in this implementation): Camilla Hollanti

Contact information for the course (valid 01.04.2021-21.12.2112):

Prof. Guillermo Mantilla-Soler: gmantelia@gmail.com, guillermo.mantillasoler@aalto.fi

TA: Taoufiq Damir : mohamed.damir@aalto.fi


CEFR level (applies in this implementation):

Language of instruction and studies (valid 01.08.2020-31.07.2022):

Teaching language: English

Languages of study attainment: English

CONTENT, ASSESSMENT AND WORKLOAD

Content
  • Applies in this implementation:

    This will be an introductory course on algebraic number theory, intended for advance undergraduates and graduate students.  In this course we will learn how tools from group theory, linear algebra, ring theory and basic analysis are usually combined to solve problems in arithmetic. Often we will use different techniques from different areas to solve the same problem and see what is the relation between such areas. For instance in the first lecture we will study the integer solutions to the Pythagorical equation 

     x 2+y2=z2.

    An integer solution of the above consists of a triple (a,b,c) in Z3 that satisfies the above equation, e.g., (3,4,5), (5,12,13) and (691, 238740, 238741). In the first lecture we will find all the integer solutions to the above using two different methods, one coming from basic geometry, and other from basic arithmetic. These two ideas will help us to motivate the study of the ring of integers of a number field and  p-adic fields. Both of these are central to the study of modern number theory. Throughout the course we will learn different methods to solve several type of equations of the above type know as Diophantine equations.




Assessment Methods and Criteria
  • Applies in this implementation:

    The final course grade will be based entirely on five homework assignments. There is no final exam.

    The passing and grading of the course is based on homework points as follows:

    There will be five weekly homework sheets, each worth 50 points (250 points total). To pass the course, you need at least 60 homework points, and minimum 10 points from each weekly homework. After having achieved this, the grade is determined (roughly) as follows:

    1: 60-100

    2: 101-140

    3: 141-180

    4: 181-220

    5: 221-250




DETAILS

Study Material

FURTHER INFORMATION

Details on the schedule
  • Applies in this implementation:

    We will cover these and probably, depending on the time, a bit more:

    • Introduction: Prehistory and history of the subject. Interesting arithmetic properties of Z e.g., unique factorization,  we know its group of units pretty well, there is an Euclidean algorithm and more. Review of basic concepts from algebra. Introducing the Gaussian integers, its properties and the concept of number field.

    • Ring of integers, Dedekind domains, fractional ideals. Ramification and the discriminant. Special attention to quadratic fields and cyclotomic extensions.

    • Failure to have unique factorization; the ideal class group. The geometry of numbers and its applications. Finiteness of the class number and Dirichlet's theorem on the unit group.

    • The Dedekind zeta function and the class number formula. A quick tour of Galois theory applied to number fields and the reciprocity law.

    • The p-adics and the Hasse-Minkowski principle.