Fields

Fields

1. What a Field Is (Ring + Inverses)

A field $(F, +, \times)$ is a commutative ring with unity where every nonzero element has a multiplicative inverse:

• $(F, +)$ abelian group; $(F \setminus \{0\}, \times)$ abelian group.
• So: $a + (-a) = 0$, $a \times a^{-1} = 1$ ($a
  e 0$); no zero divisors; cancellation always works; division (by nonzero) defined.
• Finite field = Galois field $GF(q)$, exists ⟺ $q = p^n$ (prime power). Crypto uses $GF(p)$ (Diffie–Hellman, DSA, elliptic curves mod $p$) and $GF(2^n)$ (AES $GF(2^8)$). Characteristic: $1 + 1 + \cdots = 0$ after $p$ additions ($p = 2$ for binary fields: $a + a = 0$, i.e. XOR).

2. Worked Examples

Field	Elements	Inverse example	Used in
$(\mathbb{Q}, +, \times)$	rationals	$(3/2)^{-1} = 2/3$	Theory (infinite)
$(\mathbb{Z}_7, +, \times)$ = $GF(7)$	$\{0..6\}$	$2^{-1} = 4$ ($2 \times 4 = 8 \equiv 1$); $3^{-1} = 5$	Toy DH/ElGamal sums
$GF(2) = \{0, 1\}$	bits	$1^{-1} = 1$	All binary arithmetic
$GF(2^8)$ (AES)	bytes as polynomials	$x^{-1} \bmod m(x)$ via Euclid (Module 3)	AES SubBytes/MixColumns

Non-example: $\mathbb{Z}_6$ not a field (2, 3, 4 lack inverses); $\mathbb{Z}_8$ not a field (evens fail). $\mathbb{Z}_n$ is a field ⟺ $n$ prime (proof: prime $n \Rightarrow \gcd(a,n) = 1 \Rightarrow$ Euclid gives inverse — Module 3 Topic 1).

3. Field Properties Exams Love

• $GF(p)^*$ is cyclic of order $p-1$: primitive element $g$ generates all nonzero (e.g. $3$ generates $\mathbb{Z}_7^*$: $3,2,6,4,5,1$). Discrete-log hardness lives here.
• Freshman's dream in characteristic $p$: $(a+b)^p = a^p + b^p$ (binomial coefficients $\binom{p}{k}$ vanish mod $p$) — basis of Fermat/Euler proofs and AES square-and-multiply checks.
• Polynomial roots: degree-$n$ polynomial has $\le n$ roots in a field (fails in rings: $x^2 - 1$ has 4 roots mod 8) — why Shamir secret sharing / Reed–Solomon need fields.
• Subfields: $GF(p^m) \subset GF(p^n)$ ⟺ $m \mid n$ (AES $GF(2^8)$ contains $GF(2^4)$, $GF(2^2)$, $GF(2)$ — used in composite-field S-box hardware).

4. Bridge to Modules 3–5

• Module 3 Topics 1–4: Euclid → inverses in $GF(p)$; polynomial Euclid → inverses in $GF(2^8)$.
• Module 4 Topic 2: every AES SubBytes/MixColumns step is $GF(2^8)$ arithmetic.
• Module 5: Fermat ($a^{p-1} = 1$ in $GF(p)^*$) → Euler ($a^{\phi(n)} = 1$ in $\mathbb{Z}_n^*$) → RSA ($m^{ed} = m$ via group orders).

Exam Points

• Field axioms + "$\mathbb{Z}_n$ field ⟺ $n$ prime" with Euclid justification — 5-mark core.
• Inverse numericals ($2^{-1} = 4$, $3^{-1} = 5$ mod 7) + generator trace.
• Cyclic $GF(p)^*$ + primitive element definition.
• Non-examples $\mathbb{Z}_6$, $\mathbb{Z}_8$ with failing elements named.