Group Ring Field

 

  1. Groups:

    • group is a mathematical structure that consists of a set of elements along with an operation (usually called addition or multiplication).
    • Key properties of a group:
      • Closure: The result of the operation on any two elements in the group remains within the group.
      • Associativity: The operation is associative (i.e., the order of performing the operation doesn’t matter).
      • Identity element: There exists an element (often denoted as e) such that combining it with any other element doesn’t change the other element.
      • Inverse element: Each element has an inverse (opposite) element within the group.
    • Example: The integers under addition form a group.
  2. Rings:

    • ring is a more general structure than a group. It includes both addition and multiplication operations.
    • Key properties of a ring:
      • Closure under addition and multiplication.
      • Associativity of both operations.
      • Distributive property: Multiplication distributes over addition.
    • Unlike groups, rings need not have multiplicative inverses for all elements.
    • Example: The set of integers with both addition and multiplication forms a ring.
  3. Fields:

    • field is a special type of ring where every nonzero element has a multiplicative inverse (except for 0).
    • Key properties of a field:
      • All nonzero elements have multiplicative inverses.
      • Commutativity of both addition and multiplication.
    • Fields are more structured than rings and allow division.
    • Example: The set of real numbers (excluding 0) forms a field.

In summary:

  • Every ring is a group, and every field is a ring.
  • ring has both addition and multiplication, while a group only has addition.
  • field is a ring where all nonzero elements have multiplicative inverses.

Remember, these structures play a fundamental role in abstract algebra and have applications in various mathematical and scientific contexts especially in Cryptography.

Comments

Popular posts from this blog

Number Theory CST 292 KTU IV Semester Honors Course Notes Dr Binu V P 9847390760

Sum of Squares

Quadratic Residues-Legendre and Jacobi Symbols