Module Handbook

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Course MAT-40-32-K-4

Plane Algebraic Curves; p-adic Numbers (4V+2U, 9.0 LP)

Course Type

SWS Type Course Form CP (Effort) Presence-Time / Self-Study
- K Lecture with exercise classes (V/U)
4 V Lecture 6.0 CP 56 h 124 h
2 U Exercise class (in small groups) 3.0 CP 28 h 62 h
(4V+2U) 9.0 CP 84 h 186 h


CP, Effort 9.0 CP = 270 h
Position of the semester 1 Sem. irreg. SuSe
Level [4] Bachelor (Specialization)
Language [EN] English
+ further Lecturers of the department Mathematics
Area of study [MAT-AGCA] Algebra, Geometry and Computer Algebra
Additional informations
Livecycle-State [NORM] Active


The course consists of the two parts [MAT-40-28-K-4] and [MAT-40-24-K-4].

In each summer semester at least one of the courses [MAT-40-30-K-4], [MAT-40-31-K-4] or [MAT-40-32-K-4] is offered.

Possible Study achievement

  • Verification of study performance: proof of successful participation in the exercise classes (ungraded)
  • Details of the examination (type, duration, criteria) will be announced at the beginning of the course.
    The certificate for the exercises ("Übungsschein") consists of the two (partial) certificates for the exercises in the courses [MAT-40-28-K-4] and [MAT-40-24-K-4].


Plane Algebraic Curves:

Compulsory Topics:

  • Affine and projective spaces, in particular, the projective line and the projective plane,
  • Plane algebraic curves over the complex numbers,
  • Smooth and singular points,
  • Bézout's Theorem for projective plane curves,
  • Topological genus of a curve and the genus formula,
  • Rational maps between plane curves and the Riemann-Hurwitz formula.

In addition, a selection of the following topics is covered:

  • Polar and Hessian curves,
  • Dual curves and Plücker’s formulas,
  • Linear systems and divisors on smooth curves,
  • Real projective curves,
  • Puiseux-parameterizations of plane curve singularities,
  • Invariants of plane curve singularities,
  • Elliptic curves,
  • Further aspects of plane algebraic curves.

p-adic Numbers:

  • construction of p-adic numbers,
  • p-adic integers, units,
  • p-adic topology,
  • Hensel's lemma,
  • algebraic closure,
  • Newton polygon,
  • inertia and ramification groups

Competencies / intended learning achievements

In the part "Plane Algebraic Curves", the students have learnt basic concepts of algebraic geometry by studying a class of algebraic varieties which is accessible by selected and simple methods.

In the part "p-adic Numbers", the students have studied extensions of a number system through p-adic numbers which are fundamental for number theory. In particular, they have studied their main properties and some simple applications.


Plane Algebraic Curves:

  • G. Fischer: Ebene algebraische Kurven,
  • E. Brieskorn, H. Knörrer: Plane Algebraic Curves,
  • E. Kunz: Introduction to Plane Algebraic Curves,
  • F. Kirwan: Complex Algebraic Curves,
  • R. Miranda: Algebraic Curves and Riemann Surfaces.

p-adic Numbers:

  • I.N. Stewart, D.O. Tall: Algebraic Number Theory,
  • M. Trifković: Algebraic Theory of Quadratic Numbers.


Further literature will be announced in the lecture; Exercise material is provided.


Registration for the exercise classes via the online administration system URM (

Requirements for attendance (informal)

More advanced knowledge from the courses [MAT-12-22-K-3], [MAT-12-21-K-3] and [MAT-12-26-K-3] is useful but not necessarily required.



Requirements for attendance (formal)


References to Course [MAT-40-32-K-4]

Course-Pool Name
[MAT-40-4V-KPOOL-4] Elective Courses Algebra, Geometry and Computeralgebra (4V, B.Sc.)
[MAT-40-KPOOL-4] Specialisation Algebra, Geometry and Computer Algebra (B.Sc.)