Normal closure
Encyclopedia
The term normal closure is used in two senses in mathematics
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Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
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- In group theoryGroup theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the normal closure of a subset of a groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is the smallest normal subgroupNormal subgroupIn abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
that contains the subset; see conjugate closureConjugate closureIn group theory, the conjugate closure of a subset S of a group G is the subgroup of G generated by SG, i.e. the closure of SG under the group operation, where SG is the conjugates of the elements of S:The conjugate closure of S is denoted or G.The conjugate closure of any subset S of a group G...
. - In field theoryField theory (mathematics)Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
, the normal closure of an algebraic extensionAlgebraic extensionIn abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
F/K is an extension field L of F such that L/K is normalNormal extensionIn abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]...
and L is minimal with this property. See normal extensionNormal extensionIn abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]...
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