Rings and Fields¶

Summary

From groups (one operation) to rings (two operations) and fields (with division). Why the integers are not always "enough" – and how ideal theory solves the problem of missing unique factorisation.

Prerequisites¶

Groups – Symmetry as the Language of Mathematics

1. From Groups to Rings¶

In a group we have one operation. But even the integers \(\mathbb{Z}\) have two: addition and multiplication. To capture both simultaneously, we need a richer structure – the ring.

The motivation comes directly from number theory: Fermat's Last Theorem is about the equation \(x^n + y^n = z^n\) in the integers. The proofs for \(n = 3\) and \(n = 4\) showed that sometimes one must extend the number domain – to \(\mathbb{Z}[\omega]\) or \(\mathbb{Z}[i]\). All these number domains are rings.

2. Ring Axioms and Examples¶

A ring \((R, +, \cdot)\) is a set \(R\) with two operations satisfying the following axioms:

\((R, +)\) is an abelian group (with identity element \(0\))
Multiplication is associative: \((ab)c = a(bc)\)
There is a unity element: \(1 \cdot a = a \cdot 1 = a\)
The distributive laws hold: \(a(b + c) = ab + ac\) and \((a + b)c = ac + bc\)

If additionally \(ab = ba\) for all \(a, b \in R\), the ring is called commutative.

The Most Important Examples¶

Ring	Description	Commutative?
\(\mathbb{Z}\)	Integers	✓
\(\mathbb{Z}/n\mathbb{Z}\)	Residue classes modulo \(n\)	✓
\(\mathbb{Z}[i] = \{a + bi \mid a, b \in \mathbb{Z}\}\)	Gaussian integers	✓
\(\mathbb{Z}[\omega] = \{a + b\omega \mid a, b \in \mathbb{Z}\}\)	Eisenstein integers	✓
\(\mathbb{Z}[\zeta_p]\)	Cyclotomic ring	✓
\(K[x]\)	Polynomial ring over a field \(K\)	✓
\(M_n(\mathbb{R})\)	\(n \times n\) matrices	✗ (for \(n \geq 2\))

Zero Divisors and Integral Domains¶

In \(\mathbb{Z}\) we have: if \(ab = 0\), then \(a = 0\) or \(b = 0\). This is not the case in every ring! In \(\mathbb{Z}/6\mathbb{Z}\), \(2 \cdot 3 = 6 \equiv 0\), even though neither \(2\) nor \(3\) is zero. Such elements are called zero divisors.

A commutative ring without zero divisors (other than \(0\)) is called an integral domain. The rings \(\mathbb{Z}\), \(\mathbb{Z}[i]\), \(\mathbb{Z}[\omega]\), and \(K[x]\) are integral domains; \(\mathbb{Z}/6\mathbb{Z}\) is not.

3. Ideals and Quotient Rings¶

In \(\mathbb{Z}\), divisibility is a central concept: \(3 \mid 12\), because \(12 = 3 \cdot 4\). The set of all multiples of \(3\) forms a subset \(3\mathbb{Z} = \{\ldots, -6, -3, 0, 3, 6, \ldots\}\) with special properties:

\(3\mathbb{Z}\) is closed under addition
For every \(r \in \mathbb{Z}\) and \(a \in 3\mathbb{Z}\), \(ra \in 3\mathbb{Z}\)

These properties define an ideal.

Definition. A subset \(I \subseteq R\) is called an ideal if: 1. \((I, +)\) is a subgroup of \((R, +)\) 2. For all \(r \in R\) and \(a \in I\), \(ra \in I\) and \(ar \in I\)

Ideals play the same role in rings as normal subgroups in groups: one can form quotient rings:

\[ R/I = \{r + I \mid r \in R\} \]

Example: \(\mathbb{Z}/n\mathbb{Z} = \mathbb{Z}/(n)\) is the quotient ring of \(\mathbb{Z}\) by the ideal \((n) = n\mathbb{Z}\).

Principal Ideals and Principal Ideal Domains¶

An ideal of the form \((a) = \{ra \mid r \in R\}\) (all multiples of an element) is called a principal ideal. An integral domain in which every ideal is a principal ideal is called a principal ideal domain (PID).

PID examples: \(\mathbb{Z}\), \(K[x]\) (polynomials over a field), \(\mathbb{Z}[i]\), \(\mathbb{Z}[\omega]\)

Not a PID: \(\mathbb{Z}[\sqrt{-5}]\) – here the ideal \((2, 1 + \sqrt{-5})\) has no single generator.

The crux in FLT

In a PID, unique prime factorisation holds. In \(\mathbb{Z}[\zeta_p]\) for general \(p\), this is not the case – this is precisely where Lamé's proof failed and Kummer invented ideal theory.

4. Fields¶

A field is a commutative ring in which every element \(a \neq 0\) has a multiplicative inverse: there exists \(a^{-1}\) with \(a \cdot a^{-1} = 1\).

In other words: in a field one can add, subtract, multiply and divide (except by \(0\)).

The Most Important Fields¶

Field	Description	Property
\(\mathbb{Q}\)	Rational numbers	smallest field of characteristic \(0\)
\(\mathbb{R}\)	Real numbers	complete, ordered
\(\mathbb{C}\)	Complex numbers	algebraically closed
\(\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}\)	Finite field with \(p\) elements	characteristic \(p\)
\(\mathbb{Q}_p\)	\(p\)-adic numbers	completion of \(\mathbb{Q}\)

Why is \(\mathbb{Z}/p\mathbb{Z}\) a field? Because \(p\) is prime: for \(a \not\equiv 0 \pmod{p}\), \(\gcd(a, p) = 1\), so by the extended Euclidean algorithm there exists \(b\) with \(ab \equiv 1 \pmod{p}\). By contrast, \(\mathbb{Z}/6\mathbb{Z}\) is not a field (because \(2 \cdot 3 = 0\)).

5. Field Extensions¶

A field extension is a pair \(K \subseteq L\) of fields. One writes \(L/K\) and calls \(L\) an extension of \(K\).

Algebraic Extensions¶

An element \(\alpha \in L\) is called algebraic over \(K\) if there is a polynomial \(f \in K[x]\) with \(f(\alpha) = 0\). The extension \(L/K\) is called algebraic if every element of \(L\) is algebraic over \(K\).

Examples: - \(\mathbb{Q}(\sqrt{2}) = \{a + b\sqrt{2} \mid a, b \in \mathbb{Q}\}\) – an extension of degree \(2\) - \(\mathbb{Q}(i) = \{a + bi \mid a, b \in \mathbb{Q}\}\) – also degree \(2\) - \(\mathbb{Q}(\zeta_p)\) – the \(p\)-th cyclotomic field, degree \(p - 1\)

The Degree of an Extension¶

The degree \([L : K]\) is the dimension of \(L\) as a \(K\)-vector space. It measures how "much larger" \(L\) is compared to \(K\).

Degree formula (tower law). For \(K \subseteq M \subseteq L\):

\[ [L : K] = [L : M] \cdot [M : K] \]

Example: \([\mathbb{Q}(\sqrt{2}, \sqrt{3}) : \mathbb{Q}] = [\mathbb{Q}(\sqrt{2}, \sqrt{3}) : \mathbb{Q}(\sqrt{2})] \cdot [\mathbb{Q}(\sqrt{2}) : \mathbb{Q}] = 2 \cdot 2 = 4\).

The Algebraic Closure¶

The algebraic closure \(\overline{K}\) of \(K\) is the smallest algebraically closed field containing \(K\). For example:

\(\overline{\mathbb{R}} = \mathbb{C}\) (Fundamental Theorem of Algebra)
\(\overline{\mathbb{Q}}\) is the set of all algebraic numbers – countable, but not equal to \(\mathbb{C}\)

The absolute Galois group \(G_{\mathbb{Q}} = \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) – the symmetry group of \(\overline{\mathbb{Q}}\) over \(\mathbb{Q}\) – is the central object in Wiles' proof.

6. Principal Ideal Domains and Unique Factorisation¶

Unique prime factorisation (UPF) states: every element of an integral domain can be written essentially uniquely as a product of prime elements. In \(\mathbb{Z}\), this is the Fundamental Theorem of Arithmetic: \(60 = 2^2 \cdot 3 \cdot 5\).

Theorem. In every principal ideal domain, UPF holds.

The chain of implications:

\[ \text{Euclidean} \implies \text{Principal ideal domain} \implies \text{Unique factorisation domain (UPF)} \]

Where UPF Fails¶

In \(\mathbb{Z}[\sqrt{-5}]\) we have two essentially different factorisations:

\[ 6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5}) \]

Here \(2\), \(3\), \(1 + \sqrt{-5}\), and \(1 - \sqrt{-5}\) are all irreducible, but the product has two different decompositions. UPF fails!

Kummer's Rescue: Factorise Ideals¶

Kummer's insight: even if UPF fails at the element level, it holds at the ideal level in every Dedekind domain. The ideal \((6) = (2)(3)\) has a unique decomposition into prime ideals:

\[ (6) = (2, 1 + \sqrt{-5})^2 \cdot (3, 1 + \sqrt{-5}) \cdot (3, 1 - \sqrt{-5}) \]

The class number \(h\) measures how far a ring is from being a PID: \(h = 1\) if and only if the ring is a PID. For \(\mathbb{Z}[\sqrt{-5}]\), \(h = 2\).

7. Why Rings and Fields Matter for FLT¶

The algebraic structures of this article form the backdrop for Wiles' proof:

Cyclotomic rings \(\mathbb{Z}[\zeta_p]\): Kummer's proof for regular primes uses the ideal structure of these rings.
Field extensions: Galois theory operates on field extensions – it is the bridge between equations and groups.
Finite fields \(\mathbb{F}_p\): The reduction of elliptic curves modulo \(p\) – that is, working over \(\mathbb{F}_p\) instead of \(\mathbb{Q}\) – yields the \(a_p\)-coefficients that appear in the \(L\)-series.
Local rings and deformation rings: In Wiles' proof, the rings \(R\) and \(T\) in the "\(R = T\)" theorem are local rings that parametrise families of Galois representations.

Ring theory provides the algebraic infrastructure on which the entire proof is built.