November 1st
Today I learned the definition of the Krull topology. (continue reading...)
November 2020
November 2nd
Today I learned about the regulator of a number field. (continue reading...)
November 3rd
Today I learned an interesting example of a field extension, namely $\QQ(\zeta_p)/\QQ(\zeta_p+\zeta_p^{-1}).$ (continue reading...)
November 4th
Today I learned the asymptotic formula for derangements. (continue reading...)
November 5th
Today I learned the law of quadratic reciprocity in function fields. (continue reading...)
November 6th
Today I learned a proof of the Basel problem from Fourier analysis. (continue reading...)
November 7th
Today I learned an explicit example where the naive approach of multiplying integral bases in a field composite does not necessarily yield an integral basis. (continue reading...)
November 8th
Today I learned that it's possible to do calculus over ratios of prime numbers, from here . (continue reading...)
November 9th
Today I learned that the outline for the proof that, in an extension of number fields $L/K,$ if for every unramified prime $\mf p$ we have that $f(\mf q/\mf p)$ is constant for all primes $\mf q$ over $\mf p,$ then $L/K$ is actually Galois. (continue reading...)
November 10th
Today I learned some perspectives on those $\arctan$ facts that pop every once in a while. (continue reading...)
November 11th
Today I learned the proof that splitting of $\mf p$ in a non-Galois extension $L/K$ is in bijection with cosets $H\sigma D_\mf P$ for some prime $\mf P$ over $\mf p$ in the Galois closure $M$ of $L/K,$ where $H$ is subgroup of $G=\op{Gal}(M/K)$ fixing $L,$ and $D_\mf P$ is the decomposition subgroup. (continue reading...)
November 12th
Today I learned the classification of nilpotent matrices with maximal index. (continue reading...)
November 13th
Today I learned that the inverse Galois problem holds for abelian groups, and it's actually fairly nice. (continue reading...)
November 14th
Today I learned a proof of the Fundamental Theorem of Algebra, using Galois theory as a translator. (continue reading...)
November 15th
Today I learned about parameterizing rationals on conics. (continue reading...)
November 16th
Today I learned about discrete valuation rings. (continue reading...)
November 17th
Today I learned the definition of the limit superior to prove the radius of convergence for a power series. (continue reading...)
November 18th
Today I learned the estimate for the number of ideals of bounded norm in imaginary quadratic fields. (continue reading...)
November 19th
Today I learned the group structure of the elements of norm $1$ in $\QQ(i),$ and some semblance of generalizations to number fields. (continue reading...)
November 20th
Today I learned that the group $\QQ/\ZZ$ has every finite cyclic group as a unique subgroup, and in fact no subgroup of it suffices. (continue reading...)
November 21st
Today I learned a nice perspective on completely multiplicative functions on $\ZZ^+\to\ZZ^+,$ from HMMT Guts. (continue reading...)
November 22nd
Today I learned the finish for the fact that the greedy algorithm for Egyptian fractions works. (continue reading...)
November 23rd
Today I learned the group structure for the rational points with prime-power denominator on the unit hyperbola $x^2-y^2=1.$ (continue reading...)
November 24th
Today I learned the group structure for rational points on the unit hyperbola. (continue reading...)
November 25th
Today I learned the reflection formula for the gamma function. (continue reading...)
November 26th
Today I learned the definition of a cokernel, from Vakil. (continue reading...)
November 27th
Today I learned the definition of homology groups from algebraic topology, from the Napkin. (continue reading...)
November 28th
Today I learned a proof that the only automorphism of $\RR$ is the identity. (continue reading...)
November 29th
Today I learned the Snake lemma as an example of diagram-chasing. (continue reading...)
November 30th
Today I learned that for any polynomial $P(x_1,\ldots,x_n)\in\ZZ_p[x_1,\ldots,x_n]$ has a root in $(\ZZ_p)^n$ if and only if has roots in each $(\ZZ/p^\bullet\ZZ)^n.$ (continue reading...)