# What is a field in ring theory?

Table of Contents

## What is a field in ring theory?

Definition. A field is a commutative ring with identity (1 ≠ 0) in which every non-zero element has a multiplicative inverse. Examples. The rings Q, R, C are fields.

**What is the equation for an elliptic curve?**

This equation defines an elliptic curve. y2 = x3 + Ax + B, for some constants A and B. Below is an example of such a curve. An elliptic curve over C is a compact manifold of the form C/L, where L = Z + ωZ is a lattice in the complex plane.

### What is equation type of equation?

A linear equation is an algebraic equation. In linear equation, each term is either a constant or the product of a constant and a single variable. If there are two variables, the graph of linear equation is a straight line. y = mx + c, m ≠ 0.

**What rings are used and write the calculation rules for the rings?**

Commutative Ring: If x • y = y • x holds for every x and y in the ring, then the ring is called a commutative ring. Ring with Unity: If there is a multiplicative identity element, that is an element e such that for all elements a in R, the equation e • a = a • e = a holds, then the ring is called a ring with unity.

#### What is rings in discrete mathematics?

The ring is a type of algebraic structure (R, +, .) or (R, *, .) which is used to contain non-empty set R. Sometimes, we represent R as a ring. It usually contains two binary operations that are multiplication and addition.

**What is group Law in elliptic curve?**

In short, the group law is defined for every pair of distinct points A, B. In case A = B, the fact that our elliptic curve is nonsingular tells us that there is a well-defined tangent line L at A. Either L ∩ E consists of two distinct points A, C or else L ∩ E consists of the single point A.

## Why is an elliptic curve a torus?

After adding a point at infinity to the curve on the right, we get two circles topologically. Since these parameterizing functions are doubly periodic, the elliptic curve can be identified with a period parallelogram (in fact a square in this case) with the sides glued together i.e. a torus.