Abstract Algebra II
Rings, fields and the algebra of polynomial equations: the ring axioms and the step from groups to rings, with examples from the integers, modular arithmetic, matrices, functions, the Gaussian integers and the quaternions, the elementary rules of ring arithmetic, subrings, units and Pell's equation, zero divisors, integral domains and fields with the theorem that every finite domain is a field, fields of fractions and the characteristic, ideals and quotient rings with prime and maximal ideals and the history of Kummer's ideal numbers and Dedekind's ideals, ring homomorphisms, the first isomorphism theorem and the Chinese remainder theorem, polynomial rings with the division algorithm, the factor theorem, gcds and interpolation, irreducibility tests including the rational root test, reduction modulo a prime, Gauss's lemma and Eisenstein's criterion, Euclidean domains, principal ideal domains and unique factorization domains with the Gaussian primes and the failure of unique factorization in Z[√−5], field extensions, minimal polynomials, Kronecker's construction and the tower law, ruler-and-compass constructions with Wantzel's proofs that the cube cannot be doubled nor the angle trisected and Gauss's regular 17-gon, splitting fields and the classification of finite fields with the Frobenius automorphism, and a capstone on the fundamental theorem of Galois theory and the insolvability of the quintic by radicals, with runnable Python examples throughout.
MATH-323 Year 3 3 credits 12 lessons · 4 h 32 min ▶ Video lessons coming soon
Offered by HSM University · Mathematics & Statistics
What you'll learn
- State the ring axioms, derive the elementary rules of ring arithmetic from them, and recognize rings, commutative rings and rings with unity among familiar number systems, matrices, functions and polynomials.
- Apply the subring test, compute the group of units of a ring, and identify units, zero divisors and nilpotent elements in \(\mathbb{Z}_n\), product rings and matrix rings.
- Distinguish integral domains from fields, prove that every finite integral domain is a field, and compute the characteristic of a ring.
- Define ideals, form quotient rings, and decide whether an ideal is prime or maximal using the criteria that \(R/P\) is a domain and \(R/M\) is a field.
- Use ring homomorphisms and the first isomorphism theorem to identify quotient rings, and apply the Chinese remainder theorem to split \(\mathbb{Z}_{mn}\).
- Carry out polynomial arithmetic, the division algorithm and the Euclidean algorithm over \(\mathbb{Q}\) and \(\mathbb{Z}_p\), and use the factor theorem to relate roots and linear factors.
- Test polynomials for irreducibility using degree arguments, the rational root test, reduction modulo \(p\), Eisenstein’s criterion and Gauss’s lemma.
- Explain the hierarchy Euclidean domain, principal ideal domain, unique factorization domain, and compute with the Gaussian integers, including division with remainder and factorization of primes.
- Construct simple field extensions as quotients \(F[x]/(p(x))\), find minimal polynomials, and compute degrees with the tower law.
- Prove that a constructible number has degree a power of 2 over \(\mathbb{Q}\), and use this to show that doubling the cube and trisecting a general angle are impossible.
- Construct splitting fields and finite fields \(GF(p^n)\), and describe the Frobenius map and the cyclic structure of the multiplicative group of a finite field.
- Compute Galois groups of small extensions, draw the Galois correspondence between subgroups and intermediate fields, and explain why the general quintic is not solvable by radicals.
Course content
12 lessons · 4 h 32 min total · final exam
- 1 From groups to rings 24 min · quiz
- 2 Rings, subrings and examples 23 min · quiz
- 3 Integral domains and fields 22 min · quiz
- 4 Ideals and quotient rings 22 min · quiz
- 5 Ring homomorphisms and the isomorphism theorems 23 min · quiz
- 6 Polynomial rings and the division algorithm 23 min · quiz
- 7 Irreducibility and factorization (Eisenstein, Gauss’s lemma) 22 min · quiz
- 8 Euclidean domains, PIDs and UFDs 22 min · quiz
- 9 Field extensions and degree 23 min · quiz
- 10 Constructibility: why you can’t trisect an angle 21 min · quiz
- 11 Splitting fields and finite fields 23 min · quiz
- 12 Capstone: Galois theory and the insolvability of the quintic 24 min · quiz
- E Final exam 20 questions
Requirements
Recommended background. You can start any course at any time.
Description
Rings, fields and the algebra of polynomial equations: the ring axioms and the step from groups to rings, with examples from the integers, modular arithmetic, matrices, functions, the Gaussian integers and the quaternions, the elementary rules of ring arithmetic, subrings, units and Pell's equation, zero divisors, integral domains and fields with the theorem that every finite domain is a field, fields of fractions and the characteristic, ideals and quotient rings with prime and maximal ideals and the history of Kummer's ideal numbers and Dedekind's ideals, ring homomorphisms, the first isomorphism theorem and the Chinese remainder theorem, polynomial rings with the division algorithm, the factor theorem, gcds and interpolation, irreducibility tests including the rational root test, reduction modulo a prime, Gauss's lemma and Eisenstein's criterion, Euclidean domains, principal ideal domains and unique factorization domains with the Gaussian primes and the failure of unique factorization in Z[√−5], field extensions, minimal polynomials, Kronecker's construction and the tower law, ruler-and-compass constructions with Wantzel's proofs that the cube cannot be doubled nor the angle trisected and Gauss's regular 17-gon, splitting fields and the classification of finite fields with the Frobenius automorphism, and a capstone on the fundamental theorem of Galois theory and the insolvability of the quintic by radicals, with runnable Python examples throughout.
Part of the Science stream, typically taken in year 3.