Date & Time: Monday, November 24, 2014, 14:30-15:30.
Venue: Ramanujan Hall
Title: Dedekind's theorem on splitting of primes and its converse
Speaker: S. K. Khanduja, IISER Mohali
Abstract: Let $ K = \mathbb{Q}(\theta) $ be an algebraic number field with $ f(x) $ as the minimal polynomial of the algebraic integer $ \theta $ over $ \mathbb{Q} $. Let $ p $ be a rational prime. Let \[ \bar{f}(x) = \bar{g}_{1}(x)^{e_{1}} \ldots \bar{g}_{r}(x)^{e_{r}} \] be the factorization of $ \bar{f}(x) $ as a product of powers of distinct irreducible polynomials over $ \mathbb{Z}/ p\mathbb{Z} $, with $ g_{i}(x) $ monic polynomials belonging to $ \mathbb{Z}[x] $. In 1878, Dedekind proved if $ p $ does not divide the index of the subgroup $ \mathbb{Z}[\theta] $ in $ A_{K} $, then $ pA_{K} = \wp_{1}^{e_{1}} \ldots \wp_{r}^{e_{r}} $, where $ \wp_{1}, \ldots, \wp_{r} $ are distinct prime ideals of $ A_{K} $, \wp_{i} = pA_{K} + g_{i}(\theta)A_{K} $ with residual degree of $ \wp_i/p $ equal to $ \deg {g}_{i}(x) $ for all $ i$. In 2008, we proved that converse of Dedekind's theorem holds, i.e. if for a rational prime $ p $, the decomposition of $ pA_K $ satisfies the above three properties, then $ p $ does not divide $ [A_K:\mathbb Z[\theta]] $. Dedekind also gave a simple criterion known as Dedekind Criterion to verify when $ p $ does not divide $ [A_K:\mathbb{Z}[\theta]] $. We will also discuss the Dedekind Criterion and its generalization. In 2014, we have proved the analogue of Dedekind's theorem for finite extensions of valued fields of arbitrary rank as well as of its converse.