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Quadratic Fields - Summer 2018

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Quadratic Fields - Summer 2018

Lecturer: William Hart

Assistant: Sophia Brenner

Lecture: Monday, 11:45-1:15, in 48-438 (first lecture is on April 9)

Tutorial: Friday, 11:45-1:15, in 48-438 (every two weeks, starting on April 13)

If you have any problems, questions or comments, feel free to contact me via email or drop by my office.


The investigation of Diophantine equations is a classical (and ongoing) problem in number theory. A typical example is the determination of all Pythagorean triple, that is, all triples (x,y,z)Z3<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>)</mo></mrow><mo>∈</mo><msup><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mn>3</mn></msup></mstyle></math> with x3+y3=z3<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true"><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>y</mi><mn>3</mn></msup><mo>=</mo><msup><mi>z</mi><mn>3</mn></msup></mstyle></math>, which can be done using elementary techniques. Another example would be the determination of all tuples (x,y)Z2<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><msup><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mn>2</mn></msup></mstyle></math> satisfying the equation y2=x32<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>2</mn></mstyle></math>, a problem studied 300 years ago by Fermat and Euler. Although we reduced the number of variables by 1, the problem gets much harder and elementary techniques cannot help us. Now Euler had the ingenious idea of writing the equation in the following form

x3=y2+2=(y2)(y+2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true"><msup><mi>x</mi><mn>3</mn></msup><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>=</mo><mrow><mo>(</mo><mi>y</mi><mo>-</mo><msqrt><mrow><mo>-</mo><mn>2</mn></mrow></msqrt><mo>)</mo></mrow><mrow><mo>(</mo><mi>y</mi><mo>+</mo><msqrt><mrow><mo>-</mo><mn>2</mn></mrow></msqrt><mo>)</mo></mrow></mstyle></math>

and then working in the ring Z[2]={a+b2a,bZ}<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true"><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mrow><mo>[</mo><msqrt><mrow><mo>-</mo><mn>2</mn></mrow></msqrt><mo>]</mo></mrow><mo>=</mo><mrow><mo>{</mo><mi>a</mi><mo>+</mo><mi>b</mi><msqrt><mrow><mo>-</mo><mn>2</mn></mrow></msqrt><mrow><mo>∣</mo></mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>Z</mi></mstyle><mo>}</mo></mrow></mstyle></math>, the ring of integers of the quadratic field Q(2)<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true"><mstyle mathvariant="bold"><mi>Q</mi></mstyle><mrow><mo>(</mo><msqrt><mrow><mo>-</mo><mn>2</mn></mrow></msqrt><mo>)</mo></mrow></mstyle></math>. Note that the investigation of y2=x32<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>2</mn></mstyle></math>, a problem involving only elements of Z<math xmlns="http://www.w3.org/1998/Math/MathML">Z</math>, has lead us naturally to much more involved objects. In this particular case the new objects are subrings of the quadratic number fields Q(d)<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true"><mstyle mathvariant="bold"><mi>Q</mi></mstyle><mrow><mo>(</mo><msqrt><mrow><mi>d</mi></mrow></msqrt><mo>)</mo></mrow></mstyle></math>dZ<math xmlns="http://www.w3.org/1998/Math/MathML"><mstyle displaystyle="true"><mi>d</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>Z</mi></mstyle></mstyle></math>, and the investigation of these rings will be the topic of this course. Of particular interest will be the failure of unique factorization and ways to quantify and overcome this via the ideal class group. To do this, we will employ algebraic, geometric and analytic techniques.

Credit Points and Problem Sets

There will be problem sets every two weeks, each of which typically contains four problems. To participate in the tutorial, please register at URM. To obtain a “Schein”, the following criteria have to be met:

  • Attending the tutorial (at most one tutorial may be missed without a valid reason).
  • Scoring at least 50% of the points.

For bachelor students, the “Schein” is a prerequisite for the oral exam, where you then can get 4.5 credit points. On the other hand, master students can always register for the oral exam (this is independent of the “Schein”). Without the “Schein” passing the oral exam will give only 3 credit points, while passing the oral exam with the “Schein” will give 4.5 credit points.


Basic knowledge of groups, rings and fields, which is taught for example in the course “Algebraische Strukturen”.

Lecture notes

Tommy Hofmann has previously prepared a set of written lecture notes for the course here

Any deviations from his course material will be posted here after the relevant lecture.

Lecture 5

Lecture 6

Lecture 7

Lecture 8

Correction: in class we had a theorem that said that a binary quadratic form (a, b, c) is positive definite iff d = b^2 - 4ac < 0. This should be corrected to say that d < 0 and a > 0. The case where d < 0 and a < 0 is called the negative definite case. It is not of interest to us in this course.

Problem Sessions

The problem sessions are in 48-438 on Friday, every second week, starting on Apr 14.

The problem sets are available here.


Please register for exams for this class with Frau Sternike on the 5th floor.


The material of the course is (partly) covered in the following books:

  • Mak Trifković, Algebraic Theory of Quadratic Numbers, Springer, 2013
  • Michael Artin, Algebra, Prentice-Hall, 1991 (Chapter 11)