Articles of reference request

Radius of convergence of power series

Given a meromorphic function on $\mathbb{C}$, is the radius of convergence in a regular point exactly the distance to the closest pole? As Robert Israel points out in his answer, that this is of course an upper bound by the Cauchy-Hadamard principle. Theo Buehler in the comments gives a refernce for the non obvious direction: […]

Exercises on Galois Theory

I need a source for exercises on classical Galois Theory, or to be more specific, Galois extensions of finite fields and the rationals as well as applications (solvability by radicals, for example). So far, I have worked with Tignol’s “Galois Theory of Algebraic Equations”. Any additional suggestions would be appreciated, whether it is a textbook […]

What is the best book to learn probability?

Question is quite straight… I’m not very good in this subject but need to understand at a good level.

Status of the classification of non-finitely generated abelian groups.

From the Wikipedia on abelian groups: By contrast, classification of general infinitely-generated abelian groups is far from complete. How far are we from a classification exactly? It seems like divisible groups have been classified. Which cases are left which we haven’t? What is the nature of these unknown cases that makes them so hard to […]

Self-study Real analysis Tao or Rudin?

The reference requests for analysis books have become so numerous as to blot out any usefulness they could conceivably have had. So here comes another one. Recently I’ve began to learn real analysis via Rudin. I would do all the exercises, and if I was unable to do them within a time limit (usually about […]

Math Textbooks for High School

I’m a high school student who is trying to figure out a complete course of self study for each year of high school. Is there a way to self learn grades of math without devoting too much time? For example, I was wondering whether there are suitable textbooks that have detailed explanations and progressive practice […]

A rigorous book on a First Course in linear algebra

I am just going to start linear algebra for my undergraduate course. And which is the best book available for linear algebra in terms of rigor and a book similar to calculus by Tom M Apostol (I like this book because of it’s rigorous and clear ideas, the qualities which I want). Note: I have […]

$ 2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$

Let $D = B(0,1) \subset \mathbb{C} $ a disc, $f$ holomorphic on $D$. I want to demonstrate that if $$ 2|f^{‘}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$$ then $f$ is linear. I know this is a well-known result. Where can I find the proof ?

Recommending books for introductory differential geometry

I was wondering if anyone could recommend some books for studying topics such as abstract manifolds, differential forms on manifolds, integration of differential forms, Stokes’ theorem, de Rham cohomology, Hodge star operator? Our text is A Comprehensive Introduction to Differential Geometry by Spivak, but I think this book is very difficult for a beginner to […]

Reference request: compact objects in R-Mod are precisely the finitely-presented modules?

Let $R$ be a ring. According to this MO question, the modules $M \in R\text{-Mod}$ such that $\text{Hom}(M, -)$ preserves all filtered colimits (the compact objects) are precisely the finitely-presented modules. Where can I find a proof of this?