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We present in a very concise manner fundamental (meta)mathematical domain-specific terminology needed for the Artificial Mathematical Intelligence program (Cognitive Metamathematics). Specifically, we briefly revise the notions of propositional and predicative logic, the most outstanding logical frameworks for modern mathematics (e.g., ZFC and NBG set theory, Peano arithmetic), and the notion of category and some of its derived notions. Moreover, a short description of fundamental algebraic, topological, and geometric notions is presented. For instance, the following notions are briefly introduced: (abelian) group, commutative ring with unity, localization, quotient ring, ideal, prime ideal, multiplicative ring, ring of polynomials in finitely many variables, field, field extension, group of automorphisms of a field (extension), (base for a) topological space, (ideal associated to a) algebraic set, ring of coordinates of an algebraic set, pre-sheaf and sheaf with values on the category of sets.

We consider the question of certifying that a polynomial in \({\mathbb Z}[x]\) or \({\mathbb Q}[x]\) is irreducible. Knowing that a polynomial is irreducible lets us recognise that a quotient ring is actually a field extension (equiv. that a polynomial ideal is maximal). Checking that a polynomial is irreducible by factorizing it is unsatisfactory because it requires trusting a relatively large and complicated program (whose correctness cannot easily be verified). We present a practical method for generating certificates of irreducibility which can be verified by relatively simple computations; we assume that primes and irreducibles in \({\mathbb F}_p[x]\) are self-certifying.

The present chapter is devoted to the basic theory of finite fields, including existence and uniqueness theorems as well as the main structural results. For this purpose, we also extend the fundamental material covered in Chapters 1 and 2 by proving several results on field extensions in general (in particular, in the first two sections).

Let K be an inversive difference-differential field and L a (not necessarily inversive) finitely generated difference-differential field extension of K. We consider the natural filtration of the extension L/K associated with a finite system \(\eta \) of its difference-differential generators and prove that for any intermediate difference-differential field F, the transcendence degrees of the components of the induced filtration of F are expressed by a certain numerical polynomial \(\chi _{K, F,\eta }(t)\). This polynomial is closely connected with the dimension Hilbert-type polynomial of a submodule of the module of Kahler differentials \(\varOmega _{L^{*}|K}\) where \(L^{*}\) is the inversive closure of L. We prove some properties of polynomials \(\chi _{K, F,\eta }(t)\) and use them for the study of the Krull-type dimension of the extension L/K. In the last part of the paper, we present a generalization of the obtained results to multidimensional filtrations of L/K associated with partitions of the sets of basic derivations and translations.

In this chapter, we study automorphism groups of fields and introduce Galois groups of finite field extensions. The term “Galois group” is often reserved for automorphism groups of Galois field extensions, which we define and study in Chap. 9. The terminology used in this book is very common and has several advantages in textbooks (i.e. it is easier to formulate exercises). A central result of this chapter is Artin’s lemma, which is a key result in the modern presentation of Galois theory. In the exercises, we find Galois groups of many field extensions and we use also use this theorem for various problems on field extensions and their automorphism groups.

We begin this chapter with some basic facts about vector spaces. These will be familiar (at least in the case of real vector spaces) to those readers who have studied linear algebra. We then focus our attention on the particular case of a field extension. A number of properties of field extensions are discussed. Let F be a field and \(f(x)\in F[x]\) a nonconstant polynomial. We demonstrate how to create a field extension in which f(x) splits into a product of polynomials of degree 1. This leads to a classification of all finite fields.

We now apply our knowledge of field extensions in order to answer three questions posed by the ancient Greeks.

In this chapter, we look at three aspects of the theory of abstract fields: the discovery that all finite fields are ‘Galois’ fields; Dedekind’s presentation of the ‘Galois correspondence’ between groups and field extensions; and the emergence of the concept of an abstract field.

Categorical equivalences between block algebras of finite groups - such as Morita and derived equivalences - are well-known to induce character bijections which commute with the Galois groups of field extensions. This is the motivation for attempting to realise known Morita and derived equivalences over non splitting fields. This article presents various result on the theme of descent. We start with the observation that perfect isometries induced by a virtual Morita equivalence induce isomorphisms of centers in non-split situations, and explain connections with Navarro's generalisation of the Alperin-McKay conjecture. We show that Rouquier's splendid Rickard complex for blocks with cyclic defect groups descends to the non-split case. We also prove a descent theorem for Morita equivalences with endopermutation source., Theorem 1.13 has been improved

In this chapter, we show that equations solvable by radicals are characterized by the solvability of their Galois groups. This immediately implies that general equations of degree 5 and above are not solvable by radicals. If one has a more modest goal to prove that the fifth degree general equation over a number field is not solvable by radicals, then there exists a simple argument by Nagell which only requires limited knowledge of field extensions and no knowledge of Galois theory. We consider Nagell’s proof in the exercises. This chapter further outlines Weber’s theorem on irreducible equations of prime degree (at least 5) with only two nonreal zeros, which are examples of non-solvable equations. We further discuss Galois’ classical theorem, which gives a characterization of irreducible solvable polynomials of prime degree. Both Galois’ and Weber’s results give examples of concrete unsolvable polynomials over the rational numbers. The solvability by real radicals in connection with “casus irreducibilis” is also discussed.

We consider Hilbert-type functions associated with finitely generated inversive difference field extensions and systems of algebraic difference equations in the case when the translations are assigned positive integer weights. We prove that such functions are quasi-polynomials that can be represented as alternating sums of Ehrhart quasi-polynomials of rational conic polytopes. In particular, we generalize the author’s results on difference dimension polynomials and their invariants to the case of inversive difference fields with weighted basic automorphisms.

Recall that a field extension \(\mathbb{k} \subset \mathbb{F}\) is said to be finite of degree d if \(\mathbb{F}\) has dimension d < ∞ as a vector space over \(\mathbb{k}\). We write \(\deg \mathbb{F}/\mathbb{k} = d\) in this case.

We prove a new upper bound for the existence of a differential field extension of a differential field $$K,\varDelta $$ that is compatible with a given field extension of K. In 2014, Pierce provided an upper bound in terms of lengths of certain antichain sequences of $${\text {I}\!\text {N}}^m$$ equipped with the product order. This result has had several applications to effective methods in differential algebra such as the effective differential Nullstellensatz problem. Using a new approach involving Macaulay's theorem on the Hilbert function, we produce an improved upper bound.

In this chapter, K will denote an arbitrary commutative ring with unit and \(\mathbb{k}\) an arbitrary field.

Let K be an inversive difference field, L a finitely generated inversive difference field extension of K, and F an intermediate inversive difference field of this extension. We prove the existence and establish properties and invariants of a numerical polynomial that describes the filtration of F induced by the natural filtration of the extension L/K associated with its generators. Then we introduce concepts of type and dimension of the extension L/K considering chains of its intermediate fields. Using properties of dimension polynomials of intermediate fields we obtain relationships between the type and dimension of L/K and difference birational invariants of this extension carried by its dimension polynomials. Finally, we present a generalization of the obtained results to the case of multivariate dimension polynomials associated with a given inversive difference field extension and a partition of the basic set of translations.

Using the connection between generalized eigenspaces of the multiplication family and primary components of a zero-dimensional polynomial ideal \(I\), we examine various methods for computing the primary and maximal components of \(I\) in the fifth chapter. In addition to the generic approach, i.e., the approach using a generic linear form to get a splitting endomorphism, we use random linear forms, the power of the Frobenius endomorphism over finite base fields, repeated factorization over field extensions, the calculation of the maximal components from the radical ideal, and the computation of the primary components from the maximal components. The chapter finishes with some recent insights into the calculation of the separable subalgebra.

In this paper, we generalize an approach of switching between different bases of a finite field to efficiently implement distinct stages of algebraic algorithms. We consider seven bases of finite fields supporting optimal normal bases of types 2 and 3: polynomial, optimal normal, permuted, redundant, reduced, doubled polynomial, and doubled reduced bases. With respect to fields of characteristic q = 7 we provide complexity estimates for conversion between the bases, multiplication, and exponentiation to a power \( q^{k} \), q-th root extraction. These operations are basic for inversion and exponentiation in \( GF\left( {7^{n} } \right) \). One needs a fast arithmetic in \( GF\left( {7^{n} } \right) \) for efficient computations in field extensions \( \left( {7^{2n} } \right) \), \( GF\left( {7^{3n} } \right),GF\left( {7^{6n} } \right) GF\left( {7^{14n} } \right),GF(7^{3 \times 14n} ) \) which are the core of the Tate pairing on a supersingular hyperelliptic curve of genus three. The latter serves for an efficient implementation of cryptographic protocols.

If F is a subfield of a field K, then K is said to be an extension of the field F. For \(\alpha \in K\), \(F(\alpha )\) denotes the subfield generated by \(F\cup \{\alpha \}\), and the extension \(F\subseteq F(\alpha )\) is called a simple extension of F. The element \(\alpha \) is algebraic over F if \(\dim _FF(\alpha )\) is finite. Field theory is largely a study of field extensions. A central theme of this chapter is the exposition of Galois theory, which concerns a correspondence between the poset of intermediate fields of a finite normal separable extension \(F\subseteq K\) and the poset of subgroups of \(\textit{Gal}_F(K)\), the group of automorphis ms of K which leave the subfield F fixed element-wise. A pinnacle of this theory is the famous Galois criterion for the solvability of a polynomial equation by radicals. Important side issues include the existence of normal and separable closures, the fact that trace maps for separable extensions are non-zero (needed to show that rings of integral elements are Noetherian in Chap. 9), the structure of finite fields, the Chevalley-Warning theorem, as well as Luroth’s theorem and transcendence degree. Attached are two appendices that may be of interest. One gives an account of fields with valuations, while the other gives several proofs that finite division rings are fields. There are abundant exercises.

In this chapter we describe a method for producing new rings from a given one. Of particular interest for applications is the case of a field extension of a given field.

The coGalois theory studies the correspondence between subfields of a radical field extension L∕K and subgroups of the coGalois group coG(L∕K)-the torsion of the quotient group L ×∕K ×. Its abstract version concerns a continuous action of a profinite group Γ on a discrete quasi-cyclic group A, establishing a Galois connection between closed subgroups of Γ and subgroups of the group Z 1(Γ, A) of continuous 1-cocycles. More generally, we study in the present work triples \((\varGamma,\mathfrak{G},\eta )\), where Γ is a profinite group, \(\mathfrak{G}\) is a profinite operator Γ-group, and \(\eta:\varGamma \longrightarrow \mathfrak{G}\) is a continuous 1-cocycle such that η(Γ) topologically generates \(\mathfrak{G}\). To any such triple one assigns a natural coGalois connection between closed subgroups of Γ and closed Γ-invariant normal subgroups of \(\mathfrak{G}\). In the abelian context, examples concern the coGalois theory of separable radical extensions, an additive analogue of it based on Witt calculus and higher Artin-Schreier theory, and an extension of the abstract cyclotomic framework to Galois algebras. Kneser triples and coGalois triples are investigated, and general Kneser and coGalois criteria are provided. Problems on the classification of certain finite algebraic structures arising naturally from these criteria are stated and partial solutions are given.