REFERENCE SHEET

Rings, ideals & fields

Track which arithmetic operations are available and which survive a quotient.

Before this Polynomials & roots, Homomorphisms & quotients

F[x] / (p) is a field ⇔ p is irreducible (p nonconstant)

Definitions

Commutative ring with identity

An abelian additive group with associative, commutative multiplication, a multiplicative identity, and distributivity. All rings on this sheet use this convention.

Ideal

An additive subgroup I⊆R such that ri∈I for every r∈R and i∈I. This makes multiplication of cosets well-defined.

Field

A nonzero commutative ring in which every nonzero element has a multiplicative inverse.

Key theorems

Quotient field criterion

For a commutative ring with identity, R/I is a field exactly when I is a maximal proper ideal.

Prime integer modulus

ℤ/nℤ is a field exactly when n is prime, for integers n≥2.

Notable applications

Finite arithmetic

Finite fields support algebraic error-correcting codes and exact computations.

Adjoining a root

The quotient ℝ[x]/(x²+1) realizes ℂ by making the class of x square to −1.

Work it out

Try before revealing

01 / Compute

Find the inverse of 3 in ℤ/7ℤ.

Hint

Find k with 3k≡1 mod 7.

Solution

The inverse is 5 because 3·5=15≡1 mod 7.

02 / Explain

Why is ℤ/6ℤ not a field?