December 1st
Today I learned the formalization of the intuition that $\ZZ_p$ consists of "formal power series evaluated at $p.$ (continue reading...)
December 2020
December 2nd
Today I learned that products commute with limits, categorically speaking. (continue reading...)
December 3rd
Today I learned about Hall's Marriage Theorem. (continue reading...)
December 4th
Today I learned that in all nonarchimedean fields $K$ with absolute value $|\bullet|,$ all triangles are isosceles. (continue reading...)
December 5th
Today I learned that, for any prime $p,$ every quadratic extension of $\QQ_p$ is contained in some cyclotomic field (not dependent on the quadratic extension). (continue reading...)
December 6th
Today I learned the matrix form of the fast Fourier transform. (continue reading...)
December 7th
Today I learned that all fields complete with respect to an archimedean valuation are either $\RR$ or $\CC.$ (continue reading...)
December 8th
Today I learned the definition of generalized eigenvectors, to move towards Jordan normal form sometime in my far future. (continue reading...)
December 9th
Today I learned a generalization of Hensel's lemma. (continue reading...)
December 10th
Today I learned a proof of Niven's theorem, the statement that the only $\theta$ for which $\theta$ and $\sin(\theta)$ are both rational give $\sin(\theta)\in\left\{0,\pm\frac12,\pm1\right\}.$ (continue reading...)
December 11th
Today I learned a proof of the Jordan normal form, from Terrence Tao . (continue reading...)
December 12th
Today I learned the definition of a Newton polygon. (continue reading...)
December 13th
Today I learned about Laplace integration. (continue reading...)
December 14th
Today I learned that, for $K$ the fraction field of $\mathcal O_K$ Dedekind, $\op{SL}_2(\mathcal O_K)$ acts transitively on $K\cup\{\infty\}$ as fractional linear transformations if and only if $K$ has class number 1, from Keith Conrad . (continue reading...)
December 15th
Today I learned the proof to the more general statement that orbits of elements of the projective space $K\PP^1$ under $\op{SL}_n(\mathcal O_K)$ is equal to the class number of $K.$ (continue reading...)
December 16th
Today I learned that completely splitting (for unramified primes) in an abelian extension is determined by $p\pmod N$ for some modulus $N.$ (continue reading...)
December 17th
Today I learned an application of the theory from yesterday. (continue reading...)
December 18th
Today I learned a proof for the fact that the area of a section of a parabola bounded by a line is $\frac43$ of the area of that section's midpoint triangle. (continue reading...)
December 19th
Today I learned an explicit expression for $L(1,\chi)$ with Gauss sums, which I think leads towards the quadratic class number formula and a somewhat elementary proof of Dirichlet's theorem. (continue reading...)
December 20th
Today I learned this finish of the proof of the class number formula. (continue reading...)
December 21st
Today I learned the core of the proof of the class number formula. (continue reading...)
December 22nd
Today I learned the details of the proof of the class number formula. (continue reading...)
December 23rd
Today I learned the correct context for the zeta function of the ring of integers in a function field. (continue reading...)
December 24th
Today I learned the proof of the prime number theorem for function fields using $\zeta_q.$ (continue reading...)
December 25th
Today I learned the proof for the relationship between Ford circles and kissing fractions. (continue reading...)
December 26th
Today I learned the refinement to Dirichlet's approximation theorem, from Ford circles. (continue reading...)
December 27th
Today I learned a proof of Fermat's Christmas theorem, from continued fractions. (continue reading...)
December 28th
Today I learned some $\lambda$ calculus for type theory. (continue reading...)
December 29th
Today I learned some examples of type-theoretic recursion and induction. (continue reading...)
December 30th
Today I learned more formally about proving propositions as types. (continue reading...)
December 31st
Today I learned about equality types. (continue reading...)