Abstract Algebra

What do all number-like systems have in common, and can algebra be done without numbers at all?

4 chapters4 min read5 turning points 1 open problem

Branched from: Group Theory + Algebraic Number Theory
Branched into: Representation Theory
Figures: William Rowan Hamilton, David Hilbert, Paul Gordan, Ernst Steinitz, Emmy Noether, Emil Artin, Bartel van der Waerden

In brief

Abstract algebra studies algebraic structures: rings, fields, modules and more, defined only by the rules their operations obey. Instead of solving a particular equation, it asks what follows from the rules alone. A theorem proved for all rings then holds at once for integers, polynomials, matrices and number systems no one has invented yet.

The approach grew from surprises: in 1843, a multiplication in which $ab \ne ba$ . It matured in Göttingen in the 1920s, when Emmy Noether showed that the right abstract conditions explain results that had needed long calculations. It became the language of most of twentieth-century mathematics.

Key ideas

Ring and fieldEnters 1910: A ring has addition and multiplication obeying the familiar rules, like the integers or polynomials. A field also allows division by any non-zero element, like the rationals or the real numbers.
Non-commutativityEnters 1843: Multiplication where order matters, $ab \ne ba$ , as in Hamilton's quaternions and in matrices. It is essential for describing rotations and quantum observables.
IdealEnters 1921: A subset of a ring closed under addition and under multiplication by anything in the ring. Dedekind's device from number theory became the central tool of ring theory.
Noetherian ringEnters 1921: A ring in which every ascending chain of ideals eventually stops. This one condition, identified by Noether, makes the theory of polynomial equations work.
Structure and homomorphismEnters 1930 – 1931: Study objects through the maps between them that preserve the operations. Noether's insistence on this viewpoint shaped algebra, topology and eventually category theory.

Chapter I

Numbers That Don't Commute

For a long time algebra meant manipulating numbers, and the laws of arithmetic, like $ab = ba$ , seemed beyond question. In 1843 William Rowan Hamilton broke one. He had spent years trying to build a three-dimensional analogue of the complex numbers. Walking along a canal in Dublin, he saw that it worked in four dimensions if multiplication was allowed to depend on order. He carved the rule into the stone of Broome Bridge. Quaternions showed that algebraic laws could be chosen, and by 1858 Cayley was studying matrices, which do not commute either.

Chapter II

Existence Without Construction

The second shock came from invariant theory, then a field of heroic calculation. Paul Gordan had spent years computing explicit generators for certain systems of polynomials. In 1890 David Hilbert proved that a finite set of generators always exists, in a few pages, by showing that the alternative leads to contradiction, without saying what the generators are. Gordan is said to have called it "not mathematics but theology". Within a few years the method had won, and algebra began to prefer understanding why to computing what.

Meanwhile Dedekind had introduced ideals to repair factorisation in algebraic number theory, and in 1910 Ernst Steinitz worked out the theory of fields from axioms alone.

Chapter III

A Closer Look: When Division Works, and When It Doesn't

Clock arithmetic modulo 5 uses the numbers $0, 1, 2, 3, 4$ , wrapping around at 5. Every nonzero number has a multiplicative inverse, a partner it multiplies with to give 1:

1 \times 1 = 1, \quad 2 \times 3 = 6 \equiv 1, \quad 4 \times 4 = 16 \equiv 1 .

So you can divide by anything nonzero. That makes the numbers modulo 5 a field, a number system as well-behaved for algebra as the rationals, though it has only five elements.

Modulo 6 is different. Here $2 \times 3 = 6 \equiv 0$ : two nonzero numbers multiply to zero. That breaks division. If 2 had an inverse, multiplying $2 \times 3 = 0$ by it would give $3 = 0$ . So the numbers modulo 6 form a ring but not a field. The difference is exactly that 5 is prime and 6 is not. A single abstract property ("no two nonzero elements multiply to zero, and every nonzero element is invertible") separates the systems where algebra works fully from those where it does not.

Hamilton's quaternions break a different rule. Their units satisfy $i^2 = j^2 = k^2 = ijk = -1$ , from which

ij = k, \qquad ji = -k .

Multiplication is not commutative, yet every nonzero quaternion has an inverse, so division still works (on the correct side). Quaternions form a division ring, not a field. That suits them to describing three-dimensional rotations, which also depend on order.

Abstract algebra's method is to list the rules a system obeys, find the theorems that follow from those rules alone, and then recognise the same structure wherever it recurs: in remainders, polynomials, matrices, rotations or cryptographic codes.

Chapter IV

Noether's Revolution

The decisive figure was Emmy Noether. Barred as a woman from a regular position at Göttingen, she lectured for years under Hilbert's name without pay. In 1921 she showed that the decomposition of ideals, known separately in number theory and in polynomial algebra, follows from one abstract condition: every increasing chain of ideals stops. Rings satisfying it are now called Noetherian. Calculations gave way to structure: study objects through the maps that preserve their operations. Her students, the "Noether boys", carried the style everywhere.

Bartel van der Waerden, who attended her lectures and Artin's, wrote them up as Moderne Algebra (1930–31). It presented algebra as the study of groups, rings and fields, and became the template of algebra courses worldwide. In 1933 Noether, who was Jewish, was dismissed by the Nazi government and emigrated to Bryn Mawr, where she died in 1935. Einstein called her the most significant creative mathematical genius since higher education for women began.

Her abstract methods fed directly into algebraic geometry, whose rebuilding by Zariski and Grothendieck is written in her language, and into representation theory.

Turning points5

1843REFORMULATION
Hamilton's quaternions
Walking along the Royal Canal in Dublin, William Rowan Hamilton realises that a four-dimensional number system works if multiplication is allowed to depend on order. He carves $i^2 = j^2 = k^2 = ijk = -1$ into Broome Bridge. It was the first algebra in which $ab \neq ba$ , and it showed that the laws of arithmetic were choices, not necessities.
William Rowan Hamilton
› Sources (1)
- Hamilton, W. R. (1844). On quaternions; or on a new system of imaginaries in algebra. Philosophical Magazine (3rd series) 25: 10–13.
1890PROOF
Hilbert's basis theorem
Paul Gordan had spent years computing, by enormous explicit calculations, finite sets of generating invariants for binary forms. David Hilbert proves that finite generating sets always exist for polynomial ideals, without constructing them, by a short abstract argument. Gordan reportedly exclaimed, "This is not mathematics, it is theology." Non-constructive existence proofs began to be accepted as mathematics.
David Hilbert, Paul Gordan
› Sources (1)
- Hilbert, D. (1890). Über die Theorie der algebraischen Formen. Mathematische Annalen 36: 473–534.
1910REFORMULATION
Steinitz's abstract theory of fields
Ernst Steinitz studies fields purely from their axioms. He classifies them by their characteristic, proves that every field has an algebraic closure, and organises field extensions systematically. It was the model for treating an algebraic structure axiomatically and completely.
Ernst Steinitz
› Sources (1)
- Steinitz, E. (1910). Algebraische Theorie der Körper. Journal für die reine und angewandte Mathematik 137: 167–309.
1921REFORMULATION
Emmy Noether's theory of ideals
In "Idealtheorie in Ringbereichen", Emmy Noether shows that the decomposition of ideals, known separately in number theory and in polynomial algebra, follows from a single abstract condition on chains of ideals. Long computations are replaced by structural arguments. Her Göttingen seminar, which she taught for years without a salary, spread the style across a generation.
Emmy Noether
› Sources (2)
- Noether, E. (1921). Idealtheorie in Ringbereichen. Mathematische Annalen 83: 24–66.
- Dick, A. (1981). Emmy Noether, 1882–1935. Birkhäuser.
1930 – 1931REFORMULATION
Van der Waerden's Moderne Algebra
Bartel van der Waerden, a young Dutch mathematician, writes up the lectures of Emmy Noether and Emil Artin as Moderne Algebra. Algebra is presented for the first time as the study of groups, rings and fields. The book set the curriculum of algebra for the rest of the century.
Emmy Noether, Emil Artin, Bartel van der Waerden
› Sources (1)
- van der Waerden, B. L. (1930–1931). Moderne Algebra (2 vols.). Springer.

Applications

Where it is used

Computer graphics
Quaternions rotate everything in 3D
Flight simulators, spacecraft attitude control, robots and video games represent rotations with Hamilton's quaternions. They avoid the "gimbal lock" of angle-based methods and interpolate smoothly between orientations.
› Sources (1)
- Shoemake, K. (1985). Animating rotation with quaternion curves. ACM SIGGRAPH Computer Graphics 19(3): 245–254.
Cryptography
Finite fields inside AES
The Advanced Encryption Standard, which encrypts much of the world's stored and transmitted data, does its mixing with arithmetic in the finite field of 256 elements. Abstract field theory is what makes its design analysable.
› Sources (1)
- Daemen, J. & Rijmen, V. (2002). The Design of Rijndael: AES — The Advanced Encryption Standard. Springer.

Open problems

Where the map runs out

Open

The Jacobian conjecture

Open as of 2026; many published proofs have been wrong.

Take a polynomial map from $n$ -dimensional space to itself whose Jacobian determinant (its local stretching factor) is a non-zero constant everywhere. Ott-Heinrich Keller conjectured in 1939 that such a map must have a polynomial inverse. The conjecture is unproved even for two variables.

Why it is hard: A constant Jacobian guarantees the map is invertible locally, and the problem is whether that forces a global polynomial inverse. It has been reduced to maps of degree three, but that reduction does not make it easier. Its simple statement has attracted many incorrect proofs, some by well-known mathematicians.
What resolving it unlocks: It is connected to several other open problems in algebra, including the Dixmier conjecture on differential operators, so a proof would settle a cluster of questions.

› Sources (1)

Bass, H., Connell, E. H. & Wright, D. (1982). The Jacobian conjecture: reduction of degree and formal expansion of the inverse. Bulletin of the AMS 7(2): 287–330.