Ring theory MOC

Modular arithmetic

Modular arithmetic concerns the algebraic properties of natural numbers that have been “cut off” at a certain point.

Modular congruence

The starting point of modular arithmetic are the modular congruence relations , where for two natural numbers

that is have the same remainder after euclidean division by .

Quotients

The congruence relations create new algebraic structures called the Algebraic quotient, denoted or equivalently , The properties of as a Congruence relation allow us to define

Hence, constitutes a Ring.

Quotient ring

In the case that is prime, constitutes an field, which follows from A finite integral domain is a field.

• Addition and multiplication are commutative (i.e. forms an Abelian group, forms an abelian monoid/group)
• There is an additive identity
• Additive inverse exists
• There is a multiplicative inverse
• Some elements (called units) may have a multiplicative inverse

Properties of modular arithmetic

• Fermat's little theorem
• Chinese remainder theorem

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