Course Plan 2024
Period 3
| Topic | Section in Bosch | Exercises | |
| Tu 16.01. | Definition of monoids and groups, basic calculation rules, examples | Introduction + 1.1 | |
| Th 18.01 | Subgroups, submonoids, homomorphisms, automorphisms, Carley’s theorem, conjugation | 1.1 | |
| Exercise session | |||
| Mo 22.01 | cosets, theorem of Lagrange, normal subgroups, factor/quotient group, isomorphism theorems | 1.2 | |
| Tu 23.01 | isomorphism theorems, generators of groups, cyclic groups (are all isomorphic to Z or Z/mZ), Fermat’s little theorem, classification of prime order groups | 1.2+1.3 | |
| Exercise session | GroupsRingsWeek2.pdf Download GroupsRingsWeek2.pdf | ||
| Tu 30.01 | group actions, orbit equation, orbit-stabilizer theorem, Burnside’s lemma, centralizer, center | 5.1 | |
| We 31.01 | class equation, p-Sylow groups, Sylow theorems (without proofs) |
5.1+5.2 | First home assignment available |
| Exercise session | GroupsRingsWeek3.pdf Download GroupsRingsWeek3.pdf | ||
| Mo 05.02 | consequences of Sylow theorems for finite (abelian) groups, fundamental theorem of finite abelian groups | Chapter 13.1 in Judson | |
| We 07.02 | proof of Sylow theorems | 5.2 | |
| Exercise session | GroupsRingsWeek4.pdf Download GroupsRingsWeek4.pdf | ||
| Tu 13.02 | proof of Sylow theorems continued | 5.2 | First home assignment due (before lecture) |
| We 14.02 | permutation groups |
5.3
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| Exercise session | GroupsRingsWeek5.pdf Download GroupsRingsWeek5.pdf | ||
| Mo 19.02 | rings, basic examples and calculation rules, definitions of subrings, units, (skew) fields, formal power series and polynomials |
2.0 and 2.1 beginning of Ch. 2.5 up to Prop. 2 (excl) |
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| Th 22.02 | degrees after adding and multiplying polynomials, ideals, ring homomorphisms, factor rings |
Prop. 3 in Ch. 2.5, 2.2 2.3 |
Second home assignment available (due: March 14) |
| Exercise session | GroupsRingsWeek6.pdf Download GroupsRingsWeek6.pdf | ||
| Mo 26.02 | universal property, isomorphism theorems, zero divisors, integral domains |
2.1 - 2.3 Prop. 2 and Cor. 4 in Ch. 2.5 |
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| We 28.02 |
field of fractions principal ideal domains, associatedness prime and maximal ideals
GCDs, prime elements, irreducible elements |
pages 59-60 2.2 and 2.3
Def. 4 - Prop. 6 in Section 2.4, Page 47 |
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| Exercise session | GroupsRingsWeek7.pdf Download GroupsRingsWeek7.pdf |
Period 4
| Date | Topic | Section in Bosch | Exercises |
| Tu 19.03 |
Chinese Remainder Theorem polynomial division Euclidean domains |
2.3 (Prop. 12 + Cor. 13) 2.1 (starting from Prop. 4) beginning of 2.4 |
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| Th 21.03 |
Euclidean domains are principal ideal domains extended Euclidean algorithm Unique Factorization Domains |
2.4 | |
| Exercise session | GroupsRingsWeek8.pdf Download GroupsRingsWeek8.pdf | ||
| Tu 26.03 |
principal ideal domains are unique factorization domains unique factorization in field of fractions primitive polynomials "valuations" and Gauß-Lemma |
2.4 and 2.7 | |
| We 27.03 |
prime elements in polynomial rings over unique factorization domains Gauß-Theorem: polynomial rings over unique factorization domains are unique factorization domains Eisenstein's criterion for irreducibility of polynomials |
2.7 and 2.8 |
|
| Exercise session | GroupsRingsWeek9.pdf Download GroupsRingsWeek9.pdf | ||
| Mo 08.04 | left/right modules, module homomorphisms, submodules, quotient modules, generating sets, bases | 2.9 | Third home assignment available |
| Th 11.04 |
every basis of a finitely generated free module has the same cardinality, called the rank of the module |
2.9 | |
| Exercise session | |||
| Tu 16.04 |
characteristic, prime subfield field extension, degree |
3.0, 3.1 3.2 |
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| Th 18.04 |
algebraic field extensions minimal polynomial finitely generated/simple field extensions |
3.2 |
|
| Exercise session | GroupsRingsWeek11.pdf Download GroupsRingsWeek11.pdf | ||
| Tu 23.04 |
finite field extension = finitely generated & algebraic field extension Kronecker's construction |
3.2 3.4 |
Third home assignment due (before lecture) |
| Th 25.04 |
algebraically closed fields algebraic closure fundamental theorem of algebra (without proof) |
3.4 | |
| Exercise session | GroupsRingsWeek12.pdf Download GroupsRingsWeek12.pdf | ||
| Tu 30.04 |
proof that every field K has a unique (up to K-isomorphism) algebraic closure preparations for splitting fields and finite fields |
3.4 | Fourth home assignment available |
| Th 02.05 |
splitting fields finite fields |
3.5 3.8 |
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| Exercise session | |||
| Mo 13.05 |
proof of existence and uniqueness of finite fields course summary |
3.8 | TrialTenta.pdf Download TrialTenta.pdf |
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Exercise session |
Fourth home assignment due (before lecture) |
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