What is a field in ring theory?

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.

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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.

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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.