The proof of Theorem71was done in a general context: how to shiftpoly_intro_rangeR k n s
from one ring to another, then adapt to instances of ring like Zn and Zp. This illustrates a
general approach taken by us. We could have proved the theorem by appealing directly to the properties of modular arithmetic betweenZnandZp, but we choose a ring-theoretic path using
ring homomorphism. This puts the investigation in a proper algebraic setting, and provides an example to apply our generic ring library.
§3.9 Remarks 55
We follow the revised AKS paper (Agrawal et al. [2004]) to bound the parameter k by a lower bound for the consecutive LCM. Most expositions regarding this parameter (Terence Tao [2009]; Daleson [2006]; Crandall and Pomerance [2005]), following the original AKS paper (Agrawal et al.[2002]), make use of a Chebyshev-type estimate for the product of consecutive primes. This has been discussed in Section 1.2 (page 2). These two approaches, based on different methods, are equivalent because both are effectively weak forms of the prime number theorem (seeChan and Norrish[2019b]).
Although not directly related to our approach, note that the Prime Number Theorem has been formalised byAvigad et al.[2007] in Isabelle proof assistant, and independently in HOL Light by JohnHarrison[2009]. The formalization of Chebyshev’s approach was taken up inAsperti and Ricciotti[2008] using Matita Interactive Theorem Prover.
Part II
Correctness
Part 2 is about the correctness of the AKS algorithm. The key is formulated in Part 1 as the Main Theorem. We first develop the required background theory of finite fields, then investigate the introspective relation by separating the exponents and polynomials into different sets. The interplay between these sets, together with the properties of polynomials with coefficients from a finite field, and the Pigeonhole Principle for injective map between finite sets, eventually stipulate the conclusion of the Main Theorem.
Chapter4
Advanced Algebra
This chapter develops more topics in finite fields that are crucial for the theory behind the AKS Main Theorem. Taking a geometric viewpoint, a finite field is a vector space over its subfields, with polynomial coefficients restricted to those of the subfields. The use of subfield polynomials give rise to irreducibility and adjoining roots, with a minimal polynomials for each field element. This leads to the existence and uniqueness of finite fields, which are essential for the cyclotomic factorisation of Xk − 1, where k is the AKS parameter and Xk − 1 is the introspective modulus. We shall establish the existence of a special irreducible factor from the cyclotomic factors ofXk − 1, which plays a key role in the proof of the AKS Main Theorem.
Algebra is nothing more than geometry, in words; geometry is nothing more than algebra, in pictures. — Sophie Germain (1831)
4.1
Finite Field Classification
The AKS Main Theorem, as formulate in the proof of Theorem73, is:
` FiniteFieldF ∧ |F|= char(F)⇒ ∀n k.aks_criteriaF n k⇒npower_of char(F) It is based on a finite fieldF satisfying certain criteria, theaks_criteriaF n k. Its proof depends on a special irreducible factorhofXk − 1, the modulus of introspective checks. This irreducible factor his used to form a polynomial quotient field F[X]/(h), with an element X. There are many irreducible factors ofXk − 1, but the proof, in one of its counting arguments, requires one with the multiplicative order of Xin this quotient field to be k, i.e., orderh(X) = k. The existence of such a special irreducible h requires the full theory of finite field existence and uniqueness up to isomorphism, which are the classification theorems for finite fields.
Toward this goal, we shall take a field/subfield pair, treating the field as an extension of the subfield. These ideas involve the vector spaces, a geometric picture for finite fields.
4.1.1 Vector Spaces
A vector spaceVSpaceS G f is another algebraic structure, with three components:
• a fieldS of scalars,
• an additive groupGof vectors, and
• a multiplication f taking a scalar and a vector, resulting in a vector.
Together they must satisfy the vector space axioms, which are expressed in HOL4 as: Definition 74. Axioms for a vector space.
VSpaceS G fdef= FieldS ∧ AbelianGroupG ∧(∀a v.a∈S∧v∈G⇒ f a v∈G)∧ (∀a b v.a∈S∧b∈S∧v∈G⇒ f a(f b v)= f (S.prod.opa b)v)∧ (∀v.v∈G⇒ f 1v=v)∧ (∀a u v.a∈S∧u∈G ∧v∈ G⇒ f a(G.opu v)=G.op(f a u) (f a v))∧ ∀a b v.a∈S∧b∈S∧v∈G⇒ f (a+b)v=G.op(f a v) (f b v)
Our formalisation of the theory of vector spaces follows the approach given inAxler [2015]. The library is a standalone development consisting of basis, spanning subspace, and linear inde- pendence. The dimension of a vector space over its subspace is the minimal number of vectors in a subspace basis to span the vector space. The minimal requirement ensures that such basis vectors are linear independent.
4.1.2 Subfields
A subfieldS of a field F has its carrier S ⊆ F, and itself is a field by keeping the same field additions and multiplications. The fact that multiplication is distributive over addition gives another view of the field/subfield relationship:
Theorem 75. A field is a vector space over its subfield.
` S 4F ⇒VSpaceS F.sum(×)
Proof. GivenS 4F, we can:
• identify the elements of the subfieldS as scalars.
• identity the elements of the abelian groupF.sumof fieldF as vectors.
• identify the field multiplication ∗ as the multiplication op of scalar to vector giving a vector.