Budu přednášet podobné věci jako před dvěma lety --- nebudu rozvíjet klasickou algebraickou teorii čísel,
jež se zabývá aritmetikou číselných těles (tj. konečných rozšíření tělesa
zlomků), i když i k tomu se v přednášce trochu dostaneme, ale budu se
věnovat rozmanitým výsledkům v teorii čísel (nebo i kombinatorice),
které se získají pomocí algebraických metod. Doufám, že během přednášky
nebo po ní se mi podaří dokončit učební text
, který jsem před dvěma lety začal vytvářet. Začnu FLT (Fermatovou
poslední větou) (i) pro polynomy a pak (ii) pro celá čísla pro exponent
n=3.

Přednáška se koná v pondělí ve 12:20 na chodbě KAM ve 2. patře a probíhá v angličtině. In this case, as the course is in English, I will write the overview below in English.

Exam questions: 1. Prove that there are infinitely many primes of the form p = 1 + mn. 2. Prove Wedderburn's theorem on skew fields. 3. Prove Fermat's last theorem ... for polynomials. 4. Prove the theorem of Ko Chao (if q > 3 is a prime number then x^2 - y^q = 1 has no solution in positive integers x, y). 5. Prove the Chevalley-Warning theorem and the corollary on multigraphs. 6. Prove Alon's Combinatorial Nullstellensatz and the corollary on hyperplanes.

Exam terms: please, contact me by e-mail.

Přednáška se koná v pondělí ve 12:20 na chodbě KAM ve 2. patře a probíhá v angličtině. In this case, as the course is in English, I will write the overview below in English.

Exam questions: 1. Prove that there are infinitely many primes of the form p = 1 + mn. 2. Prove Wedderburn's theorem on skew fields. 3. Prove Fermat's last theorem ... for polynomials. 4. Prove the theorem of Ko Chao (if q > 3 is a prime number then x^2 - y^q = 1 has no solution in positive integers x, y). 5. Prove the Chevalley-Warning theorem and the corollary on multigraphs. 6. Prove Alon's Combinatorial Nullstellensatz and the corollary on hyperplanes.

Exam terms: please, contact me by e-mail.

1st lecture on October 13, 2014. Contents of the course. Two applications of complex roots of unity. Cyclotomic polynomials and their properties (we need just that their coefficients are integers). 1st application: for every integer m > 0 there are infinitely many primes of the form p = 1 + mn.

2nd lecture on October 20, 2014. 2nd application: the theorem of Wedderburn that no finite skew field (=non-commutative field) exists. FLT (Fermat's last theorem). FLT for polynomials. The Stothers - Mason theorem (if f + g = h for coprime polynomials from C[t], not all constant, then max(deg f, deg g, deg h) <rad(fgh)), proof on the next lecture. Corollary: if f^n + g^n = h^n for three polynomials from C[t], not all constant, then n < 3. Formulation of the abc conjecture.

3rd lecture on October 27, 2014. Proof of the Stothers - Mason theorem. FLT for numbers. The theorem of Sophie Germain: If p = (q - 1)/2 > 2 where p and q are two primes (e.g., p = 3, 5, 11, 23, ...) then the 1st case of the FLT holds for the exponent p, that is, x^p + y^p +z^ = 0 for integers x, y, z => xyz is zero modulo p, proof.

4th lecture on November 3, 2014. FLT for exponent p = 3. Proof that x^3 + y^3 + z^3 = 0 => xyz = 0 (x, y, z are integers) according to Edward's book on the FLT. Conclusion next time.

5th lecture on November 10, 2014. Conclusion of the proof of FLT for n = 3.

November 17, 2014 - no classes, state holiday.

6th lecture on November 24, 2014. Catalan's problem (E. Catalan, 1844). Since 2004 this is P. Mihailescu's theorem: the only solution of x^m - y^n = 1 in integers x, y, m, n > 1is 3^2 - 2^3 = 1. I will prove it in the case when n = 2 or m = 2. Theorem of V. A. Lebesgue in 1850 is case n = 2: for odd m > 1the equation x^m = 1+ y^2 has no solution in nozero integers x, y, proof. The case m = 2 is more difficult and needs two lemmas. L1 says that if a, b are distinct coprime integers and p is a prime then the gcd((a^p - b^p) / (a - b), a - b) is 1 or p, and L2 says that if a, b, c, d are positive integers such that d is not a square and a^2 - d = b^2 - c^2d = 1, then for some integer n > 0 one has b + c.d^{1/2} = (a + d^{1/2})^n. I prove L2 next time. Theorem 1 (Ko Chao, 1965): If q > 3 is a prime then x^2 - y^q = 1 has no solution in positive integers x, y. We deduced Thm 1 from Theorem 2 (Nagell, 1921): If q > 2 is a prime and x, y are positive integers such that x^2 - y^q = 1 then 2 divides y and q divides x. We prove Thm 2 next time. This still leaves out to resolve x^2 - y^3 = 1. Literature: Bilu, Bugeaud, Mignotte, The Problem of Catalan, Springer, 2014.

7th lecture on December 1, 2014. Lemma 2 (on Pell equation) more generally: if a, b, c, d, e > 0 are integers, d is not a square, a^2 - b^2d = 1, c^2 - e^2d = 1, and b is minimum (in the sense that if f, g > 0 are integers with f^2 - g^2d = 1 then g >= b), then for some integer n > 0 we have c + ed^{1/2} = (a + bd^{1/2})^n, proof. Proof of Thm 2. Theorem 3 (Euler): x^2 - y^3 = 1 has only five integral solutions, (3, 2), (-3, 2), (1, 0), (-1, 0), and (0, -1). Statement of some lemmas and deduction of Thm 3 from them.

8th lecture on December 8, 2014. Proof of the 4 lemmas. The Chevalley-Warning theorem and Combinatorial Nullstellensatz. Thm (Ch.-W.): if P_1, ..., P_m are in F[x_1, ..., x_n], where char(F) = p and deg P_1 + ... + deg P_m < n, then P_1 = ... = P_m = 0 has in F kp solutions, for some k in N_0. Combinatorial application: every 4-reg. multigraph + 1 edge contains a (non-empty) 3-reg. submultigraph.

9th lecture on December 15, 2014. Proof of the Ch.-W. theorem. Alon's Combinatorial Nullstellensatz: if P is in F[x_1, ..., x_n] (char(F) is arbitrary), cx_1^{k_1}...x_n^{k_n} is a maximum degree monomial in P (i.e., c is nonzero and k_1+ ... + k_n = deg P) and A_i are n subsets of F with |A_i| > k_i, then for some elements a_i in A_i the value P(a_1, a_2, ..., a_n) is nonzero, proof. Combinatorial application: the minimum number of hyperplanes in R^n that cover all vertices of the cube {0, 1}^{{1, 2, ..., n}} but one equals n, proof.

10th lecture on January 5, 2015. Sidon sets. A set of integers X is Sidon set if all distances between two elements of X are distinct. Let S(n) be the maximum size of a Sidon subset of {1, 2, ..., n}. Theorem 1 (Erdos and Turan, 1941; Lindstrom, 1969): S(n) < n^{1/2} + n^{1/4} + O(1), proof. Theorem 2 (Erdos, 1944; Chowla, 1944): S(m^2 + m + 1) > m if m is a power of a prime number, proof.

January, 2015

2nd lecture on October 20, 2014. 2nd application: the theorem of Wedderburn that no finite skew field (=non-commutative field) exists. FLT (Fermat's last theorem). FLT for polynomials. The Stothers - Mason theorem (if f + g = h for coprime polynomials from C[t], not all constant, then max(deg f, deg g, deg h) <rad(fgh)), proof on the next lecture. Corollary: if f^n + g^n = h^n for three polynomials from C[t], not all constant, then n < 3. Formulation of the abc conjecture.

3rd lecture on October 27, 2014. Proof of the Stothers - Mason theorem. FLT for numbers. The theorem of Sophie Germain: If p = (q - 1)/2 > 2 where p and q are two primes (e.g., p = 3, 5, 11, 23, ...) then the 1st case of the FLT holds for the exponent p, that is, x^p + y^p +z^ = 0 for integers x, y, z => xyz is zero modulo p, proof.

4th lecture on November 3, 2014. FLT for exponent p = 3. Proof that x^3 + y^3 + z^3 = 0 => xyz = 0 (x, y, z are integers) according to Edward's book on the FLT. Conclusion next time.

5th lecture on November 10, 2014. Conclusion of the proof of FLT for n = 3.

November 17, 2014 - no classes, state holiday.

6th lecture on November 24, 2014. Catalan's problem (E. Catalan, 1844). Since 2004 this is P. Mihailescu's theorem: the only solution of x^m - y^n = 1 in integers x, y, m, n > 1is 3^2 - 2^3 = 1. I will prove it in the case when n = 2 or m = 2. Theorem of V. A. Lebesgue in 1850 is case n = 2: for odd m > 1the equation x^m = 1+ y^2 has no solution in nozero integers x, y, proof. The case m = 2 is more difficult and needs two lemmas. L1 says that if a, b are distinct coprime integers and p is a prime then the gcd((a^p - b^p) / (a - b), a - b) is 1 or p, and L2 says that if a, b, c, d are positive integers such that d is not a square and a^2 - d = b^2 - c^2d = 1, then for some integer n > 0 one has b + c.d^{1/2} = (a + d^{1/2})^n. I prove L2 next time. Theorem 1 (Ko Chao, 1965): If q > 3 is a prime then x^2 - y^q = 1 has no solution in positive integers x, y. We deduced Thm 1 from Theorem 2 (Nagell, 1921): If q > 2 is a prime and x, y are positive integers such that x^2 - y^q = 1 then 2 divides y and q divides x. We prove Thm 2 next time. This still leaves out to resolve x^2 - y^3 = 1. Literature: Bilu, Bugeaud, Mignotte, The Problem of Catalan, Springer, 2014.

7th lecture on December 1, 2014. Lemma 2 (on Pell equation) more generally: if a, b, c, d, e > 0 are integers, d is not a square, a^2 - b^2d = 1, c^2 - e^2d = 1, and b is minimum (in the sense that if f, g > 0 are integers with f^2 - g^2d = 1 then g >= b), then for some integer n > 0 we have c + ed^{1/2} = (a + bd^{1/2})^n, proof. Proof of Thm 2. Theorem 3 (Euler): x^2 - y^3 = 1 has only five integral solutions, (3, 2), (-3, 2), (1, 0), (-1, 0), and (0, -1). Statement of some lemmas and deduction of Thm 3 from them.

8th lecture on December 8, 2014. Proof of the 4 lemmas. The Chevalley-Warning theorem and Combinatorial Nullstellensatz. Thm (Ch.-W.): if P_1, ..., P_m are in F[x_1, ..., x_n], where char(F) = p and deg P_1 + ... + deg P_m < n, then P_1 = ... = P_m = 0 has in F kp solutions, for some k in N_0. Combinatorial application: every 4-reg. multigraph + 1 edge contains a (non-empty) 3-reg. submultigraph.

9th lecture on December 15, 2014. Proof of the Ch.-W. theorem. Alon's Combinatorial Nullstellensatz: if P is in F[x_1, ..., x_n] (char(F) is arbitrary), cx_1^{k_1}...x_n^{k_n} is a maximum degree monomial in P (i.e., c is nonzero and k_1+ ... + k_n = deg P) and A_i are n subsets of F with |A_i| > k_i, then for some elements a_i in A_i the value P(a_1, a_2, ..., a_n) is nonzero, proof. Combinatorial application: the minimum number of hyperplanes in R^n that cover all vertices of the cube {0, 1}^{{1, 2, ..., n}} but one equals n, proof.

10th lecture on January 5, 2015. Sidon sets. A set of integers X is Sidon set if all distances between two elements of X are distinct. Let S(n) be the maximum size of a Sidon subset of {1, 2, ..., n}. Theorem 1 (Erdos and Turan, 1941; Lindstrom, 1969): S(n) < n^{1/2} + n^{1/4} + O(1), proof. Theorem 2 (Erdos, 1944; Chowla, 1944): S(m^2 + m + 1) > m if m is a power of a prime number, proof.

January, 2015