October 1st
Today I learned the next lemma in the classification of finite abelian groups. (continue reading...)
October 2020
October 2nd
Today I learned the definition of the semidirect product. (continue reading...)
October 3rd
Today I learned the finish of the proof in the classification of finite abelian groups. (continue reading...)
October 4th
Today I learned an interesting application of M\"obius inversion. (continue reading...)
October 5th
Today I learned about some characters of $(\QQ,+),$ as always from Keith Conrad. (continue reading...)
October 6th
Today I learned a loose connection between Fourier analysis and theory of Dirichlet series and convolutions. (continue reading...)
October 7th
Today I learned about adeles $\AA_\QQ$ in order to construct the characters over $(\QQ,+).$ (continue reading...)
October 8th
Today I learned a nice argument that if all roots of a polynomial $P(x)\in\ZZ[x]$ have magnitude no more than $1,$ then they are all either $0$ or roots of unity. (continue reading...)
October 9th
Today I learned that every character $\chi:\QQ\to S^1$ can indeed by expressed as $\Psi_a$ for some adele $a\in\AA_\QQ.$ (continue reading...)
October 10th
Today I learned the finale to the classification of $\widehat\QQ.$ (continue reading...)
October 11th
Today I learned that the commutator subgroup (subgroup generated by $aba^{-1}b^{-1}$) of $S_n$ is $A_n,$ for $n\ge3.$ (continue reading...)
October 12th
Today I learned that the leading digits of Fibonacci numbers $F_n$ follow Benford's Law. (continue reading...)
October 13th
Today I learned that the connection between $\left(\frac\bullet p\right)$ and quadratic forms doesn't need quadratic reciprocity. (continue reading...)
October 14th
Today I learned that for a number field $K,$ there exists an extension $L/K$ such that every ideal $I\subseteq\mathcal O_K$ has $I\mathcal O_L$ principal (!), from this post . (continue reading...)
October 15th
Today I learned the proof for the existence of non-Euclidean principal ideal domains. (continue reading...)
October 16th
Today I learned the proof for the lower bound that comparison sorts must take at least $\Omega(n\log n)$ time. (continue reading...)
October 17th
Today I learned a little bit about Eisenstein's criterion, in hopes of finding a polynomial for which Eisenstein's criterion fails for any transformation. (continue reading...)
October 18th
Today I learned how to generate an irreducible polynomial which can't be proven with the standard tricks behind Eisenstein's Criterion. (continue reading...)
October 19th
Today I learned about Midy's Theorem . (continue reading...)
October 20th
Today I learned a proof for a slightly weaker version of Cohn's irreducibility criterion from Murty . (continue reading...)
October 21st
Today I learned the correct context for the statement last week that if $P(x)\in\ZZ[x]$ only has roots which have magnitude less than or equal to $1,$ then all of those roots are roots of unity. (continue reading...)
October 22nd
Today I learned that the trace pairing over finite fields gives all possible $\FF_q$-linear maps $\FF_{q^r}\to\FF_q,$ from here . (continue reading...)
October 23rd
Today I learned a reason that monogenic extensions are, in a sense, kind of rare. (continue reading...)
October 24th
Today I learned the proof of Wedderburn's Theorem showing that finite domains $D$ are all fields. (continue reading...)
October 25th
Today I learned the definition of the categorical kernel. (continue reading...)
October 26th
Today I learned an elementary proof that the set of primes which split completely in a Galois extension $K/\QQ$ of degree $n$ have density $1/n,$ only assuming some minor analytic facts about the Dedekind zeta function. (continue reading...)
October 27th
Today I learned a proof (after having skimmed one a few days ago) that a Galois extension is completely determined by its completely split primes. (continue reading...)
October 28th
Today I learned an equivalent (Vakil's) definition of a kernel in a category with $0$ objects. (continue reading...)
October 29th
Today I learned the spigot algorithm for hexadecimal digits of $\pi.$ (continue reading...)
October 30th
Today I learned a cute but unsurprising connection in some infinite Galois theory. (continue reading...)
October 31st
Today I learned the definition of Pr\"ufer groups. (continue reading...)