GALOIS THEORY - TUTORIAL 9

MISJA F.A. STEINMETZ

1. Finite Fields

Suppose that p is a prime number. Recall that Fp := Z/pZ is a field. Moreover, if K is a finite field of characteristic p then K is a finite extension of Fp. Writing [K : Fp] = r, it follows that #K = pr. Conversely, we may ask: does there exist a finite field of size pr for any r? The answer is yes.

Non-Example 1 The ring Z/prZ is not a field for any r > 1.

If this is not the way to get fields of size pr, how do we get them? For q = pr, if K has size q, then the multiplicative group K× has size q − 1. This means that αq−1 = 1 for any α ∈ K×. If we include 0 we thus see that all the elements of K satisfy the equation Xq − X. This suggests the following definition.

q Definition. Let Fq be the splitting field of X − X over Fp. Remark. Since splitting fields are only well defined up to isomorphism, this definition only defines Fq up to isomorphism. r Theorem. Suppose p is prime, q = p and Fq is as above. Then

(i) #Fq = q; (ii) any finite field of size q is isomorphic to Fq; (iii) The extension Fq/Fp is Galois and the Galois group Gal(Fq/Fp) is cyclic of order r gen- p erated by φ : Fq → Fq defined by φ : x 7→ x .

Example 2 ∼ If f(X) ∈ Fp[X] is an irreducible polynomial of degree r, then Fp[X]/(f(X)) = Fpr .

Date: 19 March 2018. 1 2 MISJA F.A. STEINMETZ

Classwork 1 4 Factorise X − X ∈ F2[X] and, hence, write an addition and multiplication table for F4 (where you can write α for one of the roots of the polynomial that are not in F2).

Classwork 2 4 Let L = F2(α) where α is a root of the irreducible polynomial f(X) = X + X + 1 ∈ F2[X]. Show that Gal(L/F2) contains exactly one subgroup H of order 2 and find β ∈ L such that H L = F2(β). GALOIS THEORY - TUTORIAL 9 3

We have looked at finite extensions K/Fp, but what about finite extensions K/Fq? Recall that this simply means that there is a homomorphism Fq → K.

Classwork 3

Prove that there exists a homomorphism Fpr → Fps if and only if s is divisible by r.

Proposition. If L/K is an extension of finite fields with #K = q, then L is Galois over K, and Gal(L/K) is cyclic, generated by the element φ : L → L defined by φ(α) = αq. Proposition. Let q = pr, where p is prime and r ≥ 1. Then the polynomial f(X) = Xq − X ∈ Fp[X] is the product of all monic irreducible polynomials in Fp[X] of degree dividing r.

Classwork 4 Prove that if p, r are primes then the number of monic irreducible polynomials of degree r r in Fp[X] is (p − p)/r. 4 MISJA F.A. STEINMETZ

Classwork 5

Find the number of irreducible monic polynomials in F2[X] of degree 6.

Strand 5.12, King’s College London E-mail address: [email protected] URL: https://www.nms.kcl.ac.uk/misja.steinmetz/