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The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize th …

Here is a proof that I think deserves to be recorded here somewhere. Of the proofs already listed it is closest to Pushkar's proof, Lucas Culler's proof, and Gian Maria Dall'Ara's highest-upvoted proo …

Here's a guess at something to try. Write $R(x) = \frac{P(x)}{Q(x)}$. Your series $F(x)$ satisfies $$F(x) = \frac{R(x)}{F(x^2)}$$ so taking logarithms we get $$\log F(x) = \log R(x) - \log F(x^2).$$ R …

I have two different and probably unrelated questions that can both be superficially described by the title, so I hope you'll forgive me if I ask them together. They both fall under the category of t …

$\zeta(s - z)$ has an Euler product $\prod_p \frac{1}{1 - p^{z-s}}$, and so a monomial $\prod_i \zeta(s - z_i)$ (with the $z_i$ not necessarily distinct) has an Euler product $$\prod_i \zeta(s - z_i) …

One of the old classic MO questions is a big-list of proofs of the fundamental theorem of algebra. Here is a second big-list question about this big list: Which of the FTA proofs can, even in prin …

You can think about tensor products as a kind of colimit; you're asking the hom functor $\text{Hom}_A(L, -)$ to commute with this colimit in the second variable, but usually the hom functor only commu …

$\sum a_n z^n$ is a rational function iff $a_n$ is a sum of polynomials times exponentials. This is a straightforward corollary of partial fraction decomposition. So, suppose $\frac{1}{2^n - 1}$ can b …

Let $C$ be the category of commutative Banach algebras and let $U : C \to \text{Set}$ be the usual forgetful functor. The holomorphic functional calculus guarantees that every holomorphic function $f …

I do not believe that the Hahn-Banach theorem is necessary. At some point I had planned on writing up a blog post verifying this but I lost the motivation... The idea is that you can prove Liouville …

Here's an idea. Group the series into blocks $$\sum_{n=dk}^{d(k+1) - 1} \frac{z^n}{n}$$ where $d$ is fixed and large enough that the complex numbers $1, z, z^2, ... z^{d-1}$ are approximately unifor …

The goal of this question is to conceptualize in some way the fact that the Riemann zeta function $\zeta(s)$, and other zeta functions like it, have analytic continuations. Background I have by now …

Let $f(x) = (1 - r_1 x)...(1 - r_n x)$ be a polynomial. Then $f(x) = 1 - e_1 x^1 + e_2 x^2 \mp ... $ where the $e_i$ are the elementary symmetric functions in the $r_i$. We define also $p_k = \sum_i …

In my answer to this question on MU, I suggested that the OP think about the difference between real-differentiable and complex-differentiable functions by using a sort of finitary analogue. One way …

If the students have had a first course in differential equations, tell them to solve the system $$x'(t) = -y(t)$$ $$y'(t) = x(t).$$ This is the equation of motion for a particle whose velocity vect …