Math · 10
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
Is the set of nonzero reals under multiplication a group? What is the identity, and what is the inverse of \(-2\)?
Timer not started. Twenty minutes of real work before a solution.
Yes. The identity is \(1\). The inverse of \(-2\) is \(-1/2\), because their product is 1. Closure, associativity, identity, and inverses all hold. Zero is excluded because it has no multiplicative inverse.
Attack 2
List the elements of \(S_3\) and find a subgroup of order 2 and a subgroup of order 3.
Timer not started. Twenty minutes of real work before a solution.
\(S_3\) has six permutations of \(\{1,2,3\}\). A subgroup of order 2 is \(\{e,(12)\}\). A subgroup of order 3 is the alternating group \(A_3=\{e,(123),(132)\}\), which is cyclic.
Attack 3
Prove that the identity element of a group is unique.
Timer not started. Twenty minutes of real work before a solution.
If \(e\) and \(f\) are both identities, then \(e=ef=f\).
Abstract algebra studies sets with operations that obey a short list of axioms, and the maps that preserve those operations. Groups capture symmetry. Rings capture arithmetic. Fields capture the setting in which linear algebra was happening all along. The payoff in a first serious course is the isomorphism theorems, the classification of finitely generated abelian groups, the Sylow theorems, and, if you go the distance, the Galois correspondence between field extensions and groups of automorphisms.
You already know linear maps are homomorphisms of additive groups that also respect scalar multiplication. This page removes the scalars and asks what is left. The work is examples: symmetric groups, cyclic groups, matrix groups, polynomial rings, and finite fields. A definition you cannot illustrate with two nonisomorphic examples is a definition you will misquote.
Judson’s Abstract Algebra: Theory and Applications is free, updated, and has enough exercises to be the whole course. Dummit and Foote is the paid encyclopedia; no edition was verified, so it is not given an ISBN here. Use it as a reference after Judson, not as a place to hide from exercises.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Groups. A group is a set with an associative binary operation, an identity, and inverses. A subgroup is a subset that is a group under the same operation. The order of an element is the size of the cyclic subgroup it generates. Lagrange: the order of a subgroup divides the order of a finite group.
Lagrange is a counting argument: the left cosets of a subgroup partition the group and all have the same size, namely the order of the subgroup.
Homomorphisms. A homomorphism preserves the operation. Its kernel is a normal subgroup, and its image is a subgroup. The first isomorphism theorem: \(G/\ker\varphi\cong\operatorname{im}\varphi\).
The kernel is the preimage of the identity. Cosets of the kernel are the fibers of the homomorphism. The induced map on cosets is a bijection onto the image and preserves the operation.
Quotients and Sylow. If \(N\) is normal in \(G\), the set of cosets \(G/N\) is a group. A finite group of order \(p^k m\) with \(p\) not dividing \(m\) has a subgroup of order \(p^k\), and the number of them is congruent to 1 mod \(p\) and divides \(m\).
The quotient operation is well-defined exactly because \(N\) is normal: \(aN\cdot bN=abN\) does not depend on representatives. The Sylow theorems are proved by letting the group act on the set of subsets of size \(p^k\), or on its Sylow subgroups by conjugation. Write the congruence \(n_p\equiv 1\pmod p\) from the class equation after you have seen one proof; do not memorize it as a spell.
Rings and fields. A commutative ring with identity has an ideal theory. Quotients by ideals are rings. A field is a commutative ring in which every nonzero element is invertible. \(F[x]\) over a field is a Euclidean domain, hence a principal ideal domain, hence a unique factorization domain.
The division algorithm in \(F[x]\) is polynomial long division. Principal ideals follow by taking a polynomial of least degree in a nonzero ideal. Unique factorization follows from the usual Euclidean-domain argument.
Galois, stated. A finite Galois extension \(K/F\) has automorphism group \(G\) with \(|G|=[K:F]\), and intermediate fields correspond bijectively to subgroups of \(G\), order-reversing. The splitting field of a separable polynomial is Galois.
This is the fundamental theorem of Galois theory. A first course often reaches it at the end. You should be able to compute the Galois group of \(x^2-2\) over \(\mathbb{Q}\) and of a cyclotomic extension of small degree before you claim the correspondence. The proof uses the theorem of the primitive element and linear independence of characters.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
Judson, the book’s front page
Read once, do not study from this. See that the book runs from groups through Galois theory and that a free PDF exists. Then download the PDF and work there.
The thing you work through.
Judson, Abstract Algebra: Theory and Applications, 2025/2026 edition PDF
Free. The HTML edition is https://judsonbooks.org/aata-files/aata-html/aata.html . Paperback ISBN 978-1-944325-21-3 is the 2026 print edition named on the author’s page. This is the main text because it is complete, exercised, and legal.
The problem book.
Judson’s harder exercises, all of them, plus the Sage checks he includes
The exercises are the problem set. Dummit and Foote is the paid source of extra problems; no edition was verified.
The hardest questions that exist at this level.
Putnam problems on groups, rings, and polynomials
Borisov’s sorted list points at the algebra problems: https://people.math.binghamton.edu/borisov/Putnam/SelectedProblemList.pdf . Qualifying exams in algebra are the real ceiling; a public one you can start now is a Putnam algebra problem you cannot start in five minutes.
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: written proofs of Lagrange, the first isomorphism theorem, and one Sylow counting argument for a specific order, plus a complete computation of the Galois group of \(x^3-2\) over \(\mathbb{Q}\) or a written admission of exactly which lemma you are missing. Fail if “normal” means “invariant” without a sentence about conjugates or kernels.
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
Define a group, prove Lagrange, and state the first isomorphism theorem with the kernel in the sentence. No notes.