Abstract Algebra II

A field extension L/K (also written K ⊆ L) occurs when a field L contains a smaller field K, making L a vector space over K; the dimension of this vector space is called the degree [L:K]. Extensions built by adjoining a root of an irreducible polynomial, K(α), are the basic building blocks, and the tower law ([L:K] = [L:M][M:K] for K ⊆ M ⊆ L) lets degrees be computed by breaking an extension into stages. This machinery, developed by Galois and later formalized by Dedekind and Artin, is the language in which classical questions like 'can an angle be trisected with straightedge and compass' get precise answers.

A finite field (also called a Galois field, denoted GF(q) or 𝔽_q) is a field with finitely many elements. Évariste Galois showed such fields exist precisely when the number of elements q is a prime power pⁿ, and for each prime power there is exactly one finite field up to isomorphism. Finite fields underlie error-correcting codes (Reed–Solomon codes), the Advanced Encryption Standard (AES uses arithmetic in GF(2⁸)), and elliptic curve cryptography, largely because their multiplicative group is always cyclic, giving predictable, well-understood algebraic structure.

A polynomial f(x) over a field F is irreducible if it cannot be factored into two polynomials of lower positive degree with coefficients in F — the polynomial analogue of a prime number. Irreducibility depends heavily on the base field: x²+1 is irreducible over ℝ but factors as (x−i)(x+i) over ℂ. Irreducible polynomials are exactly the ones for which F[x]/(f(x)) is a field, making them the essential ingredient for constructing field extensions, finite fields, and (via Eisenstein's criterion and reduction mod p) for proving polynomials cannot be factored over ℚ.

The splitting field of a polynomial f(x) over a field K is the smallest field extension of K in which f factors completely into linear factors. Splitting fields exist for every polynomial over every field and are unique up to isomorphism, and they are the natural home for Galois theory, since the Galois group of f is defined as the automorphism group of its splitting field fixing K. Constructing a splitting field is an iterative process: adjoin one root at a time via a quotient K[x]/(irreducible factor), and repeat until the polynomial fully factors.

A field is a commutative ring in which every nonzero element has a multiplicative inverse, so addition, subtraction, multiplication, and division (by nonzero elements) all behave the way they do for rational, real, or complex numbers. Fields are the algebraic setting for linear algebra (vector spaces need a field of scalars) and for Galois theory (which studies field extensions).

Galois theory studies field extensions by associating to each extension a group of symmetries, the Galois group, that permutes the roots of a polynomial while fixing the base field. The celebrated Fundamental Theorem of Galois Theory translates questions about intermediate fields into questions about subgroups, and this translation is what finally proved that no general formula using radicals can solve quintic (degree 5) or higher polynomial equations.

Because F[x] is a Euclidean domain (and hence a unique factorization domain) whenever F is a field, every nonzero polynomial over F factors uniquely, up to order and unit scalars, into a product of irreducible polynomials over F. This mirrors unique factorization of integers into primes, with irreducible polynomials playing the role of primes. What changes dramatically from one field to another is which polynomials count as irreducible: the same polynomial can be irreducible over ℚ, split into distinct linear factors over ℝ or ℂ, or factor in yet another pattern over a finite field 𝔽_p. Comparing a polynomial's factorization pattern across ℚ, ℝ, ℂ, and 𝔽_p (via reduction mod p) is one of the most practical tools for proving irreducibility and for computing Galois groups.

The vector space axioms never actually require the scalars to be real numbers — they work verbatim for scalars drawn from any field F, whether that's ℚ, ℂ, a finite field 𝔽_p, or an abstract field arising in Galois theory. This generalization is not a mere curiosity: every field extension L/K is automatically a vector space over K (with the same field addition and multiplication restricted to scalars in K), and the degree [L:K] used throughout field theory is defined exactly as the dimension of this vector space. Working over general fields also produces genuinely new phenomena absent over ℝ or ℂ, such as finite-dimensional spaces with only finitely many vectors total (over 𝔽_p) and fields of positive characteristic where familiar facts about linear independence must be re-derived from the axioms rather than geometric intuition.

The Galois group Gal(L/K) of a field extension L/K is the group of all field automorphisms of L that fix every element of K, under composition. When L is the splitting field of a polynomial f over K, Gal(L/K) acts faithfully by permuting the roots of f, so it can always be realized as a subgroup of a symmetric group S_n on the n roots — this concrete permutation picture is what makes Galois groups computable in practice. Identifying which subgroup of S_n a given polynomial's Galois group actually is (cyclic, dihedral, or the full S_n) is one of the central computational problems of Galois theory, since the group's structure directly answers questions about solvability by radicals and the shape of the subfield lattice.

A polynomial equation is solvable by radicals if its roots can be expressed starting from the coefficients using only the field operations (+, −, ×, ÷) together with taking nth roots, exactly the way the quadratic formula solves ax²+bx+c=0 using a single square root. Galois's landmark theorem translates this analytic-looking question into a purely group-theoretic one: a polynomial is solvable by radicals over a field of characteristic 0 if and only if its Galois group is a solvable group — one built up from abelian pieces via a subnormal series. Since the symmetric group S₅ (and every Sₙ for n ≥ 5) is not solvable, this immediately proves the Abel–Ruffini theorem: there is no general algebraic formula, analogous to the quadratic formula, for the roots of a general polynomial of degree 5 or higher.

The tensor product V ⊗_F W of two vector spaces over a field F is the universal home for bilinear maps: rather than building a new space directly, it is defined by the property that every bilinear map out of V × W factors uniquely through a single linear map out of V ⊗ W. This 'linearize the bilinear' trick converts multilinear problems into linear-algebra problems, underlies the definition of tensors in physics and differential geometry, lets scalars be extended (base change) from F to a bigger field, and generalizes cleanly from vector spaces to modules over a ring, where it becomes one of the central constructions of commutative algebra.

An algebraic number is any complex number that is a root of some nonzero polynomial with rational coefficients; an algebraic integer is a root of a monic polynomial with integer coefficients. Every algebraic integer is an algebraic number, but not conversely (1/2 is algebraic — root of 2x−1 — but not an algebraic integer). The algebraic integers form a ring, generalizing ℤ, and this ring inside a number field is the central object of algebraic number theory, playing the role that ℤ plays inside ℚ. Numbers that are not algebraic at all, like π and e, are called transcendental, and their existence (proved by Liouville in 1844 and Hermite/Lindemann later in the 19th century) settled ancient questions like squaring the circle.

In any commutative ring of prime characteristic p, the map sending x to xᵖ is a ring homomorphism — a fact that looks like a coincidence until you notice it follows from the binomial theorem collapsing modulo p. This map, called the Frobenius endomorphism after Ferdinand Georg Frobenius, is the single most important tool for studying finite fields: on 𝔽_q with q = pⁿ, it generates the entire Galois group of 𝔽_q over its prime subfield 𝔽_p, and its fixed points are exactly the elements of 𝔽_p. Because it converts the potentially hard problem of finding roots of unity or automorphisms into simple exponentiation, the Frobenius map underlies primality tests, the structure theory of Galois groups over finite fields, and constructions in algebraic number theory (Frobenius elements) and algebraic geometry (Frobenius morphisms on varieties over 𝔽_p).

An integral domain is a commutative ring with unity 1 ≠ 0 that has no zero divisors: if a·b = 0, then a = 0 or b = 0. This single condition is what allows 'canceling' in equations (ab = ac with a ≠ 0 implies b = c) exactly as with ordinary integers, which is where the name comes from. The integers ℤ are the archetypal example; every field is automatically an integral domain, and integral domains are precisely the rings that embed into a field of fractions.

A module over a ring R generalizes the notion of a vector space by allowing the 'scalars' to come from a ring instead of a field — an R-module is an abelian group M equipped with an action of R that respects addition and the ring structure. Because rings need not have multiplicative inverses, modules can behave very differently from vector spaces: they need not have a basis, and two modules of 'the same size' need not be isomorphic. Every abelian group is a ℤ-module, and modules over F[x] classify linear operators on vector spaces, making module theory a unifying language across algebra.

Given a ring R and an ideal I, the quotient ring R/I is the set of cosets of I, made into a ring by (a+I)+(b+I) = (a+b)+I and (a+I)(b+I) = (ab)+I. Ideals play the role in ring theory that normal subgroups play in group theory — they are exactly the substructures by which you can quotient and still get a ring, because absorption under multiplication guarantees the coset product is well-defined. The most familiar example is ℤ/nℤ, and quotients like F[x]/(p(x)) for irreducible p(x) are the standard way to construct field extensions.

A ring homomorphism is a function between two rings that preserves both addition and multiplication (and typically the multiplicative identity), the ring-theoretic analogue of a group homomorphism. Its kernel — the elements mapping to zero — is always an ideal of the domain, and by the First Isomorphism Theorem for rings, the quotient by the kernel is isomorphic to the image. Ring homomorphisms are the structure-preserving maps used to compare rings, define quotient constructions, and evaluate polynomials at specific points.

A unique factorization domain (UFD) is an integral domain in which every nonzero non-unit element factors into irreducible elements, and that factorization is unique up to reordering and multiplication by units — exactly generalizing the Fundamental Theorem of Arithmetic for ℤ. The ring of integers ℤ and any polynomial ring F[x] over a field F are UFDs, and Gauss's lemma shows that if R is a UFD, so is the polynomial ring R[x]. Not every integral domain is a UFD, however: the classic counterexample is ℤ[√-5], where 6 factors two genuinely different ways into irreducibles.

An ideal is a special subset of a ring that absorbs multiplication by any ring element, playing the same role for rings that normal subgroups play for groups: ideals are precisely the kernels of ring homomorphisms and the objects used to form quotient rings.

A polynomial ring R[x] is built by taking all polynomials with coefficients from a ring R and treating them as a new ring under polynomial addition and multiplication. Polynomial rings are the setting for factorization, root-finding, and the construction of field extensions, and they inherit many properties of their coefficient ring R.

A ring is a set equipped with two operations, addition and multiplication, that generalizes the arithmetic of integers: addition makes the set an abelian group, multiplication is associative, and multiplication distributes over addition. Rings capture the essential structure shared by ℤ, polynomial systems, and matrix algebras, without requiring multiplicative inverses.

Maximal and prime ideals single out the two most important flavors of proper ideal in a commutative ring, and both are best understood through the quotient rings they produce. An ideal M is maximal when no proper ideal sits strictly between M and the whole ring R — equivalently, R/M is a field. An ideal P is prime when it satisfies the ring-theoretic analogue of 'a prime number divides a product only if it divides a factor' — equivalently, R/P is an integral domain. Since every field is an integral domain, every maximal ideal is automatically prime, but the converse can fail. These two ideal types organize the algebraic geometry of Spec(R) and are the engine behind unique factorization, localization, and the structure theory of commutative rings.

A ring R is Noetherian, named after Emmy Noether, if its ideals cannot form an infinitely increasing chain — every ascending sequence of ideals I₁ ⊆ I₂ ⊆ I₃ ⊆ ... must eventually stabilize. This single finiteness condition, equivalent to every ideal being finitely generated, is what makes most of the rings used in practice (ℤ, polynomial rings over a field, and any of their quotients) tame enough to support induction arguments, the Hilbert basis theorem, and primary decomposition. Noether isolated this property in the 1920s precisely because it is the weakest hypothesis under which 'ideal theory' behaves the way it does for ℤ.

Modules over a principal ideal domain (PID) — rings like ℤ or K[x] where every ideal is generated by one element — enjoy a structure theory almost as clean as vector spaces. The Structure Theorem for Finitely Generated Modules over a PID says every such module splits as a direct sum of a free part and finitely many cyclic torsion pieces, each a quotient R/(p^k) for a prime p. Specializing R = ℤ recovers the classification of finite abelian groups; specializing R = K[x] recovers Jordan/rational canonical form for a linear operator. This single theorem is therefore the common ancestor of two results usually taught separately.

Algebraic geometry studies geometric objects — varieties — defined as solution sets of polynomial equations. Classical algebraic geometry works over algebraically closed fields (especially \(\mathbb{C}\)); modern algebraic geometry (after Grothendieck) works with schemes, sheaves, and cohomology over arbitrary rings. The field unifies algebra, geometry, topology, and number theory.

Schemes are the central objects of modern algebraic geometry, introduced by Grothendieck in the late 1950s to provide a unified framework for studying algebraic varieties over arbitrary fields and rings — not just algebraically closed fields. A scheme is a locally ringed space locally isomorphic to the spectrum of a commutative ring. This generality allows techniques from number theory and commutative algebra to be applied geometrically, and underlies the proof of the Weil conjectures, the Langlands program, and modern arithmetic geometry.

Chain complexes and exact sequences are the central combinatorial and algebraic machinery of homological algebra. A chain complex organizes modules and linear maps into a sequence where consecutive compositions vanish; its homology measures the 'holes' or 'obstructions'. Exact sequences are the special case where there are no holes at all. The long exact sequence of a short exact sequence of complexes is one of the most powerful tools in algebra and topology.

Ext and Tor are the derived functors of Hom and tensor product respectively. They measure the failure of these functors to be exact. Ext classifies module extensions and encodes obstruction theory; Tor detects torsion and measures how far a module is from being flat. Together they are the main computational tools of homological algebra.

Homological algebra is the branch of mathematics that uses algebraic techniques — particularly chain complexes, exact sequences, and derived functors — to study algebraic structures. It arose from algebraic topology (computing homology groups of spaces) and now pervades algebra, geometry, and representation theory. Its core insight is that many algebraic properties can be measured by the failure of certain functors to be exact.