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groups ring fields in advanced mathematics​

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A group ring field is a ring that is defined over a group and a field. It is a ring that is also a field, meaning that every element in the ring has a multiplicative inverse.

Some examples of group ring fields include:

  • The group ring field of integers over the field of rational numbers, denoted as Z/Q.
  • The group ring field of polynomials over the field of real numbers, denoted as R[x].
  • The group ring field of matrices over the field of complex numbers, denoted as M(n,C).

Group ring fields have applications in algebra, representation theory, and number theory. They are used to study algebraic structures that arise from the combination of group theory and field theory.

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