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0x010 Abstract Algebra

Group

Definition (group) A group is an ordered pair \((G, *)\) where \(G\) is a set and \(*\) is a binary operation on \(G\) satisfying following axioms

  • associative: \((a*b)*c = a*(b*c)\)
  • identity: \((\exists e \in G) (\forall a \in G) a*e = e*a = a\)
  • inverse: \((\forall a \in G) (\exists a^{-1} \in G) a*a^{-1} = a^{-1}*a = e\)

Definition (Abelian) The group is called abelian if \(a*b = b*a\)

Subgroup

Theorem (Lagrange) If \(H\) is a subgroup of a group \(G\), then

\[\left|G\right|=\left[G:H\right]\cdot \left|H\right|\]

Ring

Modules

Field

Definition (field) A field is a set \(F\) together with two binary operations called addition and multiplication, it has to satisfy the following axioms

  • associative: \(a+(b+c) = (a+b)+c, a*(b*c) = (a*b)*c\)
  • commutative: \(a+b=b+a, a*b=b*a\)
  • identity: \(a+0=a, a*1=a\)
  • inverse: \(a+(-a)=0, a*a^{-1}=1\)
  • distributive: \(a*(b+c)=(a*b)+(a*c)\)

Galois

Reference

[1] Dummit, David Steven, and Richard M. Foote. Abstract algebra. Vol. 3. Hoboken: Wiley, 2004.