Integral Domains and Fields

Tom Judson · GFDL · Abstract Algebra: Theory and Applications, Ch. 7

An integral domain is a commutative ring with no zero divisors: if ab = 0 then a = 0 or b = 0. A field is an integral domain where every nonzero element has a multiplicative inverse. Fields are where you can divide freely. Every finite integral domain is a field.

Zero divisors

In Z6, we have 2 * 3 = 0. Neither 2 nor 3 is zero, so they are zero divisors. Zero divisors break cancellation: if ab = ac and a is a zero divisor, you cannot conclude b = c. Integral domains ban zero divisors to keep cancellation safe.

Scheme

; Zero divisors in Z6
; Find all pairs where a*b = 0 mod 6 with a,b nonzero

(display "Zero divisors in Z6:") (newline)
(let loop ((a 1))
  (when (< a 6)
    (let inner ((b 1))
      (when (< b 6)
        (when (= 0 (modulo (* a b) 6))
          (display "  ") (display a) (display " * ")
          (display b) (display " = 0") (newline))
        (inner (+ b 1))))
    (loop (+ a 1))))

; Z5 has no zero divisors (5 is prime)
(newline) (display "Zero divisors in Z5:") (newline)
(let loop ((a 1))
  (when (< a 5)
    (let inner ((b 1))
      (when (< b 5)
        (when (= 0 (modulo (* a b) 5))
          (display "  ") (display a) (display " * ")
          (display b) (display " = 0") (newline))
        (inner (+ b 1))))
    (loop (+ a 1))))
(display "  (none)")

Fields

A field is a commutative ring where every nonzero element has a multiplicative inverse. Z5 is a field: every element 1,2,3,4 has an inverse mod 5. Z6 is not: 2 and 3 have no inverses. Z_p is a field if and only if p is prime.

Scheme

; Find multiplicative inverses in Z5

(define (find-inverse a n)
  (let loop ((b 1))
    (cond
      ((= b n) #f)
      ((= 1 (modulo (* a b) n)) b)
      (else (loop (+ b 1))))))

(display "Inverses in Z5:") (newline)
(let loop ((a 1))
  (when (< a 5)
    (display "  ") (display a) (display "^-1 = ")
    (display (find-inverse a 5)) (newline)
    (loop (+ a 1))))

(newline) (display "Inverses in Z6:") (newline)
(let loop ((a 1))
  (when (< a 6)
    (let ((inv (find-inverse a 6)))
      (display "  ") (display a) (display "^-1 = ")
      (display (if inv inv "none")) (newline))
    (loop (+ a 1))))
; 2 and 3 have no inverses in Z6.

Containment hierarchy

Fields are the most structured. Every field is an integral domain (no zero divisors). Every integral domain is a commutative ring. Every commutative ring is a ring.

Polynomial rings

Given a ring R, the polynomial ring R[x] consists of polynomials with coefficients in R. If R is an integral domain, so is R[x]. Polynomial rings over fields are particularly well-behaved: you can do polynomial long division.

Scheme

; Polynomials over Z5 (a field)
; Represent polynomial as list of coefficients: (a0 a1 a2 ...)
; p(x) = 2 + 3x + x^2 -> (2 3 1)

(define (poly-eval coeffs x n)
  (let loop ((cs (reverse coeffs)) (result 0))
    (if (null? cs)
        result
        (loop (cdr cs)
              (modulo (+ (car cs) (* result x)) n)))))

(define p '(2 3 1))  ; 2 + 3x + x^2

(display "p(x) = 2 + 3x + x^2 over Z5") (newline)
(let loop ((x 0))
  (when (< x 5)
    (display "  p(") (display x) (display ") = ")
    (display (poly-eval p x 5)) (newline)
    (loop (+ x 1))))

Notation reference

Math	Scheme	Python	Meaning
ab = 0 ⇒ a=0 or b=0	no zero divisors	no zero divisors	Integral domain
a⁻¹	(find-inverse a n)	find_inv(a, n)	Multiplicative inverse
Z_p	(modulo (* a b) p)	(a*b)%p	Field when p is prime
R[x]	(poly-eval coeffs x n)	poly_eval(coeffs,x,n)	Polynomial ring over R

Neighbors

Algebra track

← Ch.6 Rings — fields are rings with multiplicative inverses
Ch.8 Lattices and Boolean Algebras — a different kind of algebraic structure
# Number Theory Ch.7 — primitive roots come from the multiplicative group of a field
🔐 Cryptography Ch.7 — elliptic curves form fields, used directly in crypto
🐱 Category Theory Ch.24 — Lawvere theories unify rings, groups, and fields

∞ Lebl Ch.1 Real Numbers — the reals as the motivating example of a complete ordered field

Translation notes

Judson covers integral domains in Ch. 16 alongside rings and fields. The fact that every finite integral domain is a field is proved via the pigeonhole principle: left-multiplication by a nonzero element is injective (no zero divisors), hence surjective (finite set), so 1 is in the image. Polynomial rings appear in Ch. 17. The containment diagram is the key mental model: more axioms, fewer examples, stronger theorems.

← Rings by june.kim Lattices and Boolean Algebras · 8 of 8 →