Modern Algebra, by Thomas Judson.
A less terse textbook with more elementary examles.
Click here for a pdf copy.
Topics in Algebra, by I. N. Herstein.
A traditional testbook for math major classes in
abstract algebra.
Beautiful classic sources
Galois Theory, by Emil Artin.
Lectures Delivered at the University of Notre Dame in 1947.
This is a beautiful and historic version of Galios theory
by the person who invted the approach.
It is a classic.
Emil Artin is the father of Michael Artin.
A Course in Arithmetic, by Jean Pierre-Serre.
Some topics in number theory including p-adic numbers
and the Hasse Minkowski theorem.
Serre is a master expositor who makes everything seem
clear and simple.
"Arithmetic" is a mistranslation of the French title,
which refers to number theory, not what English speakers
call arithmetic.
The first parts are a great reference for finite fields and
quadratic reciprocity.
Click here for a pdf copy.
Number Theory, by Z. I. Borevich and I. R. Shafarevich.
Similar topics to the Serre book, but beautifully motivated
with examples and special cases.
Linear Representations of Finite Groups, by Jean Pierre-Serre.
Group representation theory is a powerful interaction
between group theory and linear algebra.
The first section of this book is "elementary" (if you
are in abstract algebra) and memorable.
Click here for a pdf copy.
The link should work for any registered NYU student.
More advanced books
Algebra, by Thomas Hungerford.
An easier graduate level textbook many students find
helpful. Look for a copy online.
Algebra, by Serge Lang.
Classic graduate level textbook on all of algebra.
Though loved by instructors, I find the book confusing
and unmotivated.
Fields and Galois Theory, by J. S. Milne.
This is a clear and professional description of the class
material on Galois theory, at a somewhat higher level
than an undergraduate textbook. Look here for helpful examples.
Milne has an amazing set of online materials on all
aspects of algebra.
https://www.jmilne.org/math/CourseNotes/FT.pdf
Articles of interest
A Probabilistic Algorithm for Testing Primality, by Michael Rabin.
Click here for a pdf copy.
Ideas related to Fermat's little theorem can be used to
test whether a given n is prime, without finding a
divisor if n is not prime!
This Miller Rabin test made possible trapdoor cryptography,
which is used for all secure web credit card transactions.
Chebotarev and his Density Theorem, by P. Stevenhagen and H. W. Lenstra.
Click here for a pdf copy.
This theorem asks about the Galois group of an integer polynomial
mod p, as a function of p.
The theorem brings in ideas related to the Riemann zeta
function.
The article outlines the interesting life of Cheboratev.