B. Sury
ISI Bangalore
July 2, 2025
Some Manifestations of How Group Theory Aids Number Theory: Is there a polynomial of degree $2025$ over the integers which is irreducible, but becomes reducible modulo each prime? How about $2026, 2027$? If $f,g$ are polynomials with integer coefficients which take same set of values modulo every prime, what is the relation between $f$ and $g$? If $f$ is an irreducible polynomial over the integers occurring as the characteristic polynomial of two matrices with integer entries, are the matrices similar over $\mathbb{Q}$? Over $\mathbb{Z}$? If $f_1, \cdots , f_r, g_1, \cdots , g_s$ are polynomials over $\mathbb{C}$ such that their compositions $f_1(f_2(....(f_r))) = g_1(g_2(...g_s)))$, what can be said about these decompositions? Given two integer polynomials $f$ and $g$, are there infinitely many integers $x,y$ such that $f(x)=g(y)$? Which integers are sums of two cubes of rational numbers? What is the size of the partition function $p(n)$? Does the Dedekind zeta function of a number field determine the field? The modus operandi used to address these diverse questions is the exploitation of a group that appears naturally in each situation. We give a panoramic view of this fascinating world.
ISI Bangalore
July 2, 2025
Some Manifestations of How Group Theory Aids Number Theory: Is there a polynomial of degree $2025$ over the integers which is irreducible, but becomes reducible modulo each prime? How about $2026, 2027$? If $f,g$ are polynomials with integer coefficients which take same set of values modulo every prime, what is the relation between $f$ and $g$? If $f$ is an irreducible polynomial over the integers occurring as the characteristic polynomial of two matrices with integer entries, are the matrices similar over $\mathbb{Q}$? Over $\mathbb{Z}$? If $f_1, \cdots , f_r, g_1, \cdots , g_s$ are polynomials over $\mathbb{C}$ such that their compositions $f_1(f_2(....(f_r))) = g_1(g_2(...g_s)))$, what can be said about these decompositions? Given two integer polynomials $f$ and $g$, are there infinitely many integers $x,y$ such that $f(x)=g(y)$? Which integers are sums of two cubes of rational numbers? What is the size of the partition function $p(n)$? Does the Dedekind zeta function of a number field determine the field? The modus operandi used to address these diverse questions is the exploitation of a group that appears naturally in each situation. We give a panoramic view of this fascinating world.