In this post, we consider the quest of computing the 5-adic expansion of 1/2. We begin with introducing p-adic integers and numbers, and discussing when certain polynomials with coefficients in the integers have zeroes in the p-adic integers. This question is closely related to Hensel's lemma, which can be proven using an algebraic version of Newton's iteration. We use this to compute approximations of rational numbers in the p-adics, and consider which p-adic numers have an eventually periodic expansion.
My name is Felix Fontein and I'm currently a software developer at Dybuster AG. Until February 2014, I used to be a postdoctoral fellow at the University of Zurich. I'm working in the area of Computational Number Theory, in particular on arithmetic in global fields, the infrastructure of such fields, computation of regulators and fundamental units, and related areas. I studied at University of Oldenburg and at the University of Zurich, and was a postdoctoral fellow at the University of Calgary before going back to Zurich.
About This Blog.
This blog focusses on my research as well as other mathematical topics which I am interested in.
- Partial Fractions.
- The Probability That Two Numbers Are Coprime.
- The Power of the Distributive Law.
- A Cute Identity.
- Solving Certain Linear Systems over the Integers.
- On a Certain Determinant.
- A Strange Inequality.
- Multiplicity of the Determinant.
- Rigorous Arithmetic in the Arakelov Divisor Class Group of a Number Field.
baby steps Cayley-Hamliton Chinese Remainder Theorem determinant DLP f-representation finite abelian group function field genus giant steps group functors Hasse derivative Hensel's lemma inequality of arithmetic and geometric mean infrastructure Jordan decomposition linear system of equations Leibniz rule Muirhead's inequality n-dimensional number field one-dimensional probability of being coprime Parseval's identity reduction Rao-Blackwell Theorem tensor product topological argument Taylor's formula universal property