GF(p), GF(p^n) construction, and the Frobenius map
There is a field with 4 elements and one with 27, but none with 6. Build finite fields from scratch and watch the map x → xᵖ shuffle their elements.
Choose a prime and switch between the addition and multiplication tables. Watch the powers of the primitive element g sweep through every nonzero element.
The primitive element g = 2 generates all 4 nonzero elements by repeated multiplication.
A finite field (or Galois field) GF(q) exists if and only if q is a prime power pⁿ. For each prime power there is exactly one finite field (up to isomorphism). GF(p) = Z/pZ is the simplest case. For n > 1, GF(pⁿ) is constructed by quotienting Fₚ[x] by an irreducible polynomial of degree n.
The field GF(p) is simply Z/pZ: integers modulo a prime. Its multiplicative group GF(p)* is cyclic of order p-1, meaning some element g generates all nonzero elements as powers g, g², ..., gᵖ⁻¹ = 1. This generator is called a primitive element.
Key insight: The multiplicative group of any finite field is cyclic. This fundamental fact underlies discrete logarithm cryptography and the construction of extension fields.
Pick a field to see the irreducible polynomial that builds it and its arithmetic tables. Check that every nonzero element has an inverse.
Elements are polynomials of degree < 2 with coefficients in GF(2). Cell colors encode the product index. Every nonzero element has a multiplicative inverse.
To build GF(pⁿ), take an irreducible polynomial of degree n over GF(p) and form the quotient GF(p)[x]/(irreducible). The elements are polynomials of degree < n, and arithmetic is performed modulo the irreducible. Different irreducible polynomials give isomorphic fields.
Key insight: GF(4) is NOT Z/4Z. It is F₂[x]/(x²+x+1), a field with four elements where every nonzero element has an inverse. Z/4Z has zero divisors (2 × 2 = 0) and is not a field.
Pick a field and hover over an element to highlight its orbit under x → xᵖ. The outlined fixed points are the prime field GF(p).
Fixed points: 2 = |GF(2)| (the prime subfield)
Frobenius order: 2
Gal(GF(22)/GF(2)) = ⟨Frob⟩, order 2
The map φ: x → xᵖ is an automorphism of GF(pⁿ) that fixes GF(p) pointwise. The Frobenius generates the Galois group Gal(GF(pⁿ)/GF(p)) ≅ Z/nZ. Its orbits reveal the subfield structure and connect finite field theory to Galois theory.
Key insight: The Frobenius automorphism has order n, and its fixed field is exactly GF(p). The subfields of GF(pⁿ) correspond to divisors of n, matching the subgroups of the cyclic Galois group.
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