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Abstract algebra

· 2 min read

(One day we will get to Reed-Solomon error correction)

Functions:

  • injective – one-to-one, hence f(x1) = f(x2) implies x1 = x2
  • surjective – onto, visi y turi bent vieną x
  • bijective – injective surjective

Relations:

  • Transitive
  • Symmetric
  • Reflective

Group (follows group theory)​

5-polynomial proof

group1 youtube

group2 youtube

Group (G, *) is a nonempty set G together with binary operation ‘*’ on G such that:

  • CLOSURE: a*b is a uniquely defined element of G.
  • ASSOCIATIVITY: a*(bc) = (ab)*c.
  • IDENTITY: ∃e ∈ G: ea = a and ae = a, ∀a ∈ G.
  • INVERSE ELEMENT: For each a ∈ G, ∃a-1 ∈ G: a-1a = e and aa-1 = e.
  • Commutative group (Abelian group): ab = ba.

Ring​

ring youtube

Ring (R, +, _) is a set R, together with two operations (+, _) which has the following properties:

  • (R, +) is an ABELIAN GROUP.
  • (R, *) has CLOSURE and ASSOCIATIVITY.
  • DISTRIBUTIVITY: a*(b+c) = (ab)+(ac) and (a+b)c = (ac)+(b*c)
  • Commutative ring: (R, *) has commutativity.

Field​

Field extensions, Galois field, arithmetic, getting ready for error correction

Field (F, +, _) is a set F together with two operations + and _, which has the following properties:

  • (F, +, *) is a COMMUTATIVE RING
  • (F, *) has INDENTITY
  • (F, *) has INVERSE

In other words – field axioms:

  • CLOSURE under * and +
  • ASSOCIATIVITY under * and +
  • COMMUTATIVITY under * and +
  • INVERSES of * and +
  • IDENTITIES of * and +
  • DISTRIBUTIVITY of * over +

Galois field (finite field) youtube

Polynomial fields