Numbers, Polynomials and Groups

Year 1 · 6 sections · 94 concepts · 382 practice questions

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What's covered

free1 · Integers and divisibility21 concepts

Overview 1 · The plan of this section
Definition 1.1 · Divisibility
Proposition 1.2 · Basic properties of divisibility
Theorem 1.3 · The division algorithm
Definition 1.4 · Greatest common divisor
Lemma 1.5 · gcd(a, b) = gcd(b, a − qb)
Theorem 1.6 · Euclid's algorithm
Theorem 1.7 · Bézout's identity
Corollary 1.8 · Every common divisor divides the gcd
Definition 1.9 · Coprime integers
Corollary 1.10 · Coprime iff ax + by = 1
Proposition 1.11 · If a | bc and gcd(a, b) = 1 then a | c
Definition 1.12 · Primes
Proposition 1.13 · Every n ≥ 2 is a product of primes
Theorem 1.14 · Euclid's lemma
Theorem 1.15 · The fundamental theorem of arithmetic
Definition 1.16 · Least common multiple
Proposition 1.17 · gcd and lcm from factorisations
Theorem 1.18 · Linear Diophantine equations
Corollary 1.19 · √n is irrational unless n is a perfect square
Check 1.20 · Check: Euclid and Bézout by hand

2 · Modular arithmetic15 concepts

Proposition 2.1 · Congruences and the integers modulo n
Definition 2.2 · Units and inverses modulo n
Theorem 2.3 · a is a unit mod n iff gcd(a, n) = 1
Definition 2.4 · Fields
Corollary 2.5 · ℤₙ is a field if and only if n is prime
Proposition 2.6 · Cancellation in congruences
Theorem 2.7 · Solving linear congruences
Theorem 2.8 · The Chinese remainder theorem
Theorem 2.9 · Fermat's little theorem
Proposition 2.10 · Euler's totient function
Theorem 2.11 · Euler's theorem
Theorem 2.12 · Wilson's theorem
Technique 2.13 · Fast modular exponentiation
Example 2.14 · RSA encryption
Check 2.15 · Check: solving congruences

3 · Complex numbers14 concepts

Definition 3.1 · Complex numbers
Proposition 3.2 · ℂ is a field
Proposition 3.3 · Conjugate and modulus
Definition 3.4 · The Argand diagram, argument and polar form
Proposition 3.5 · Multiplication in polar form
Definition 3.6 · The exponential form e^{iθ}
Theorem 3.7 · De Moivre's theorem
Corollary 3.8 · Multiple-angle and power formulas
Theorem 3.9 · Roots of unity
Proposition 3.10 · Sums of roots of unity
Theorem 3.11 · nth roots of a complex number
Proposition 3.12 · Geometry in the complex plane
Example 3.13 · Quadratic equations over ℂ
Check 3.14 · Check: working with complex numbers

4 · Polynomials14 concepts

Definition 4.1 · Polynomials over a field
Theorem 4.2 · Division algorithm for polynomials
Theorem 4.3 · The remainder theorem
Corollary 4.4 · The factor theorem
Theorem 4.5 · A polynomial of degree n has at most n roots
Proposition 4.6 · Multiple roots and the derivative
Theorem 4.7 · gcds of polynomials and Bézout
Theorem 4.8 · Irreducible polynomials and unique factorisation
Theorem 4.9 · The fundamental theorem of algebra
Proposition 4.10 · Real polynomials and conjugate roots
Theorem 4.11 · Vieta's formulas
Proposition 4.12 · The rational root test
Proposition 4.13 · Lagrange interpolation
Check 4.14 · Check: factorising a polynomial

5 · Permutations12 concepts

Definition 5.1 · Permutations and the symmetric group
Proposition 5.2 · S_n has n! elements
Definition 5.3 · Cycles
Theorem 5.4 · Disjoint cycle decomposition
Proposition 5.5 · Disjoint cycles commute
Proposition 5.6 · The order of a permutation
Proposition 5.7 · Permutations are products of transpositions
Theorem 5.8 · The sign of a permutation
Corollary 5.9 · Sign from the cycle type
Definition 5.10 · The alternating group
Proposition 5.11 · Conjugating a permutation relabels its cycles
Check 5.12 · Check: computing with permutations

6 · Groups18 concepts

Definition 6.1 · Groups
Example 6.2 · Standard examples of groups
Proposition 6.3 · Basic properties of groups
Definition 6.4 · The dihedral groups
Proposition 6.5 · The order of an element
Theorem 6.6 · Subgroups and the subgroup test
Proposition 6.7 · The cyclic subgroup generated by an element
Theorem 6.8 · Subgroups of cyclic groups are cyclic
Proposition 6.9 · Orders of elements in a cyclic group
Lemma 6.10 · Cosets partition the group
Theorem 6.11 · Lagrange's theorem
Corollary 6.12 · Consequences of Lagrange's theorem
Definition 6.13 · Homomorphisms, kernel and image
Proposition 6.14 · Basic properties of homomorphisms
Proposition 6.15 · Isomorphisms
Proposition 6.16 · Direct products and ℤₘ × ℤₙ
Theorem 6.17 · Groups of order at most 5
Check 6.18 · Check: is it a group, a subgroup, an isomorphism?

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Proposition 1.2 · Basic properties of divisibility

Divisibility is transitive and survives integer linear combinations: $a\mid b, a\mid c\Rightarrow a\mid bx + cy$ . Size: a divisor of $b\ne0$ is at most $|b|$ in size. Mutual divisibility: $a = \pm b$ . Why it matters: (ii) is used constantly: anything dividing $a$ and $b$ divides every $ax + by$ .

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