Algebraická teorie čísel / Algebraic Number Theory, should be more correctly and precisely: Algebraic Methods
in Number Theory and Combinatorics, NDMI066, winter term 2024/25
Friday 17:20-18:50 in S6 (on November 29 elsewhere)
This time I will lecture on P. Mihailescu's proof of Catalan's conjecture and I will use the
booklet by R. Schoof, as well as the other two
books mentioned below.
Catalan's conjecture is nicely summarized by the very first page of
the book
by P. Ribenboim (which is from the era when Catalan's conjecture was
still a conjecture and not a theorem): CATALAN'S 3^2 - 2^3=1! X^U - Y^V=1? CONJECTURE Are 8 and 9 the Only Consecutive Powers?
The third
book
on Catalan's conjecture is due to Yu. F. Bilu, Y. Bugeaud and M. Mignotte. You see what an exciting and interesting
problem/theorem it is. I restarted my Diophantine Tetralogy project and begin to write its first
part (state by November 1, 2024).
In it I will cover, besides other things, my lectures in the course, with the exception of lecture 1.
Lecture 1 on October 4, 2024. lecture 1 (more or less). The Catalan problem and
a very rough outline of Mihailescu's proof.
Lecture 2 on October 11, 2024. Chapter 2 of my text The equation x^m - y^2 = 1 for odd
m > 1 has no nonzero solution - the theorem of V. Lebesgue (1850).
Lecture 3 on October 18, 2024. Chapters 3.1 and 3.2 of my text. My 1989 solution of the equation
x^2 - y^3 = 1, all solutions are (+-3, 2), (+-1, 0) and (0, -1).
Lecture 4 on October 25, 2024. Again Chapters 3.1 and 3.2 of my text. My 1989 solution of the equation
x^2 - y^3 = 1.
Lecture 5 on November 1, 2024. Chapter C.1 of my text. UFDs, Euclidean domains and
a proof of the theorem that Euclidean => UFD.
Lecture 6 on November 8, 2024. Chapter 4.1 of my text. Solving the Diophantine equation x^2 - y^q = 1,
q > 3 is a prime number - the theorem of Chao Ko (1965) and its proof by Chein (1976).
Lecture 7 on November 15, 2024. Chapter 5 of my text (more or less complete). The relations of Cassels
(1953, 1960): if x, y is a nonzero solution of x^p - y^q = 1, where p, q > 2 are primes, then p divides y and q divides x.
Lecture 8 on November 22, 2024. Conclusion of the proof of the harder relation of Cassels. Proof of the lemma on binom(a/q, k).
Lecture 9 on November 29, 2024. On number fields (mostly after Appendix A of the BBM book). See Appendix D.1 (to be written)
of my text
Lecture 10 on December 6, 2024. T. 1. There are oo many primes. L. 2. An integer n has a prime divisor iff n
differs from -1 and 1. Proof of T. 1. L. 3. (1) Every n>0 is a product of primes and (2) if p divides n (in N) then n is a product
of primes involving p. T. 4. (FTAr) Every n in N is a unique product of primes. Review of Euclidean domains. D. 5. Definition
of an Euclidean domain. T. 6. Every Euclidean domain is UFD - we proved this earlier. C. 7. Z[i] is an Euclidean domain -
exercise for you. D. 8. Definition of a principal ideal domain (PID). T. 9. Every Euclidean domain is PID. L. 10. If I
is an ideal (in a ring R) then I = R iff 1_R is in I. Prime ideals and maximal ideals. See C1, D1 and D2 of
my text.
Lecture 11 on December 13, 2024. P. 10,5. Let R be a ring and I be a nonzero ideal in it. Then R/I is a domain iff I is prime,
and R/I is a field iff I is maximal. Axiom 11. Zorn's lemma: If (X, <) is a poset in which every chain has an upper bound, then for
every x in X there is an element y in X that dominates x and is maximal in <; this axiom is equivalent with the axiom of choice
C. 12. Every ideal in any ring is a subset of a maximal ideal. C. 13 (of P. 10,5). In any ring every maximal ideal is
a prime ideal. P. 13,5. In every PID every nonempty set of ideals has an inclusion-wise maximal element. T. 14.
Every PID (principal ideal domain) is UFD (unique factorization domain).
Lecture 12 on December 20, 2024. Proof of T. 14. T. 15. If a = (1 + sqrt{-19})/2 then the domain Z[a] (a subring of C) is
PID but not Euclidean; without proof. D. 16. Noetherian rings. P. 17. In any ring R it is equivalent that (i) R is Noetherian,
(ii) every nonempty set of ideals in R has an inclusion-wise maximal element, and (iii) there is no infinite strictly increasig chain
of ideals in R. Radical of an ideal and primary ideals. P. 18. Radical of an ideal is an ideal. P. 19. Radical of a primary
ideal is a prime ideal. T. 20.
The Lasker--E. Noether theorem; without proof. Remarks about E. Lasker
and W. Steinitz.
Lecture 13 on January 10, 2025.
Exam questions (final form) 1. The theorem of V. Lebesgue (1850) (Chapter 2 of
my text). 2. x^4 - 3y^2 = 1 has no nontrivial solution (Theorem 3.1.6 of my text).
3. Prove that every Euclidean domain is UFD (Theorem D.1.11 of my text or you may follow
(your notes of) my presentation in lecture 5). 4. Prove that x^2 - y^q = 1, q > 3 is a prime number, has no nontrivial solution
(Chapter 4.1 of my text). 5. Prove that if p>q>2 are primes and nonzero integers x, y
satisfy that x^p - y^q=1, then q divides x. (Chapter 5.1 of my text). 6. Prove that every
PID is UFD (Chapter D.2 of my text or you may follow (your notes of) my presentation in lecture 12).
December 2024