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A finite field has q = pⁿ elements for a prime p and positive integer n. For each prime power q there exists a unique finite field up to isomorphism. The Frobenius automorphism x ↦ xᵖ generates the Galois group of F_q over F_p.
From your work on splitting fields, you know how to build field extensions by adjoining roots of polynomials. Finite fields are the payoff of that machinery applied over the simplest base fields, the fields ℤ/pℤ = 𝔽_p with p elements. The central question is: what finite fields can exist, and how many? The answer turns out to be strikingly clean. A finite field must have prime-power order. You can see why from basic group theory: the multiplicative group 𝔽* has order |𝔽| - 1, and the additive group has order |𝔽|. The additive group has characteristic p for some prime, so |𝔽| = pⁿ. Conversely, for every prime power q = pⁿ, a field with exactly q elements exists and is unique up to isomorphism.
The construction uses the splitting field you already know how to build. The field 𝔽_{pⁿ} (also written GF(pⁿ)) is the splitting field of the polynomial xpⁿ − x over 𝔽_p. Why this polynomial? Because over 𝔽_{pⁿ}, every element α satisfies α^{pⁿ} = α — this follows from Fermat's little theorem lifted to the multiplicative group, which has order pⁿ − 1. So the pⁿ elements of 𝔽_{pⁿ} are precisely the roots of xpⁿ − x, which has exactly pⁿ distinct roots (its derivative is −1, so it has no repeated roots). This is why the construction works: we get the field we want as the root set of a specific polynomial.
The uniqueness statement has real bite. It means there is no ambiguity — if you and a colleague both construct "the field with 16 elements" by different methods (say, you adjoin a root of x⁴ + x + 1 over 𝔽_2, and they adjoin a root of x⁴ + x³ + 1), you will get isomorphic fields. There is essentially one 𝔽_{16}. This contrasts sharply with rings, where non-isomorphic rings of the same size are common, and it is a special feature of fields that arithmetic imposes such strong rigidity.
The Frobenius automorphism φ: x ↦ xᵖ is the key symmetry of finite fields. It is a field automorphism because (a + b)ᵖ = aᵖ + bᵖ in characteristic p (the binomial coefficients C(p,k) for 0 < k < p are all divisible by p, so those terms vanish). Applying φ repeatedly gives φ² : x ↦ xp², φ³ : x ↦ xp³, and so on. After n applications, φⁿ : x ↦ xpⁿ = x, which is the identity on 𝔽_{pⁿ}. So φ has order exactly n, and it generates the entire Galois group Gal(𝔽_{pⁿ}/𝔽_p) ≅ ℤ/nℤ. The Galois group of a finite field extension is always cyclic, always generated by Frobenius — a beautiful rigidity that makes finite fields far more tractable than general field extensions.