Title: On a Theorem of Ore
Speaker: Sudesh K. Khanduja (Indian Institute of Science)
Abstract: PDF. Let $K=\mathbb Q(\theta)$ be an algebraic number field with $\theta$ in the ring $A_K$ of algebraic integers of $K$ and $F(x)$ be the minimal polynomial of $\theta$ over rational numbers. For a prime $p,$ let $\overline F(x)=\overline g_1(x)^{e_1}\cdots\overline g_r(x)^{e_r}$ be the factorization of the polynomial $\overline F(x)$ obtained by reducing coefficients of $F(x)$ modulo $p$ into a product of powers of distinct irreducible polynomials over $\mathbb Z/p\mathbb Z$ with each $g_i(x)\in \mathbb Z[X]$ monic. The determination of the prime ideal decomposition in $A_{K}$ of any rational prime $p$ is one of the important problems in Algebraic Number Theory. In 1894, Hensel developed a powerful approach by showing that the prime ideals of $A_{K}$ lying over $p$ are in one-to-one correspondence with the monic irreducible factors of $F(x)$ over the field $\mathbb {Q}_p$ of $p$-adic numbers and that the ramification index together with the residual degree of such a prime ideal can also be determined from these irreducible factors. Hensel's Lemma leads to the factorization $F(x)=F_1(x)\cdots F_r(x)$ over the ring $\mathbb {Z}_p$ of $p$-adic integers with $\overline{F_{i}}(x)=\overline {g_i}(x)^{e_i}.$ If $p$ divides $[A_K:\mathbb Z[\theta]],$ then these factors $F_i(x)$ need not be irreducible over $\mathbb Q_p.$ In 1928, Ore attempted to determine further decomposition of $F_i(x)$ into a product of irreducible factors over $\mathbb Q_p$ using the $g_i$-Newton polygon of $F_i(x)$ for each $i.$ In 2012, we extended the scope of Ore's Theorem when the base field is an arbitrary field $K$ with a real valuation $v$ which is not necessarily discrete and deduced an analogue of Ore's Theorem for Dedekind domains (see [ On prolongations of valuations via Newton polygons and liftings of polynomials, {\it J. Pure Appl. Algebra} {\bf{216}} (2012) 2648--2656]). In 2014, we extended this theorem to henselian valued fields of arbitrary rank. As an application, we have derived the analogue of Dedekind's Theorem regarding splitting of rational primes in algebraic number fields as well as of its converse for general valued fields extending similar results proved for discrete valued fields in [On a Theorem of Dedekind, {\it Int.J. Num.Theory} {\bf{4}}(2008) 1019-1025].. We have also given a reformulation of Hensel's Lemma for polynomials with co-efficients in henselian valued fields which is used in the proof of extended Ore's Theorem and was proved in [On irreducible factors of polynomials over complete fields, {\it J. Algebra Appl.} {\bf{12}} (2013) 1250125] in the particular case of complete rank one valued fields.