September 8th
Today I learned the definition of perfect fields, from Keith Conrad : $K$ is perfect if and only if all irreducible polynomials $K[x]$ are separable. (continue reading...)
September 2020
September 9th
Today I learned the definition of the different. (continue reading...)
September 10th
Today I learned the definition of "uniformly continuous'' from one of Tom's riddles: show that any uniformly continuous function defined on a bounded domain is bounded. (continue reading...)
September 11th
Today I learned some facts about the different ideal $\diff I,$ where $I$ is a fractional ideal of $\mathcal O_K.$ (continue reading...)
September 12th
Today I learned that $\FF_q$ is also a perfect field for prime-powers $q,$ continuing from the Conrad expository paper. (continue reading...)
September 13th
Today I learned that the Cayley-Hamilton Theorem can sometimes be used as a lemma (!) instead of just always a theorem, from one of Tom's riddles. (continue reading...)
September 14th
Today I learned a more geometric interpretation of the $A^*$ in the definition of the different ideal. (continue reading...)
September 15th
Today I learned the definition and some examples of the functor of points, which I think is my prototypical example of a contravariant functor. (continue reading...)
September 16th
Today I learned the definition of localization of a ring. (continue reading...)
September 17th
Today I learned a very natural proof of Lucas's Theorem: for $p$ prime and integers $m=\sum_km_kp^k$ and $n=\sum_kn_kp^k$ written in base $p,$ we have\[\binom mn\equiv\prod_{k=0}^\infty\binom{m_k}{n_k}\pmod p.\] (continue reading...)
September 18th
Today I learned the categorical definition of the direct sum/coproduct. (continue reading...)
September 19th
Today I learned an example a proof by universal properties, by proving that localization commutes with arbitrary direct sums. (continue reading...)
September 20th
Today I learned about fibered products, which are basically products that make diagrams commute. (continue reading...)
September 21st
Today I learned some applications of the idea that prime-splitting can tell us useful information about Galois groups. (continue reading...)
September 22nd
Today I learned about Yoneda's Lemma. (continue reading...)
September 23rd
Today I learned that the relative different is kind of multiplicative in towers, which unsurprisingly follows roughly from the behavior of trace in towers, with some fudging around. (continue reading...)
September 24th
Today I learned the definition of a filtered poset for the categorical colimit. (continue reading...)
September 25th
Today I learned a proof that the relative different perfectly encodes ramification information of an extension $L/K.$ (continue reading...)
September 26th
Today I learned the sequence construction for the inverse limit in $\texttt{Set}$, or $\texttt{Grp}$ or $\texttt{Rings}$ or $\texttt{Vec}_k$ for that matter. (continue reading...)
September 27th
Today I learned why the Solovay–Strassen (probable) primality test can give confidence of $\left(\frac12\right)^{\text{witnesses}}.$ (continue reading...)
September 28th
Today I learned the definition of ramification groups. (continue reading...)
September 29th
Today I learned a construction giving a construction for a witness to the Solovay–Strassen primality test, from Keith Conrad . (continue reading...)
September 30th
Today I learned a somewhat easy precursor to the classification of finite abelian groups; it wasn't even as hard as it felt it would be in my head. (continue reading...)