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The subgroup generated by a single element, that is, the closure of this element, is called a cyclic group. In linear algebra, the closure of a non-empty subset of a vector space (under vector-space operations, that is, addition and scalar multiplication) is the linear span of this subset.
For any two integers and , the sum + is also an integer; this closure property says that + is a binary operation on . The following properties of integer addition serve as a model for the group axioms in the definition below.
The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a. Next we will prove the base case b = 1, that 1 commutes with everything, i.e. for all natural numbers a, we have a + 1 = 1 + a.
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
A linear subspace is a vector space for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied. [11] The closure property also implies that every intersection of linear subspaces is a linear subspace. [11] Linear span
In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x.One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.
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In this regard, the algebraic closure of F q, is exceptionally simple. It is the union of the finite fields containing F q (the ones of order q n). For any algebraically closed field F of characteristic 0, the algebraic closure of the field F((t)) of Laurent series is the field of Puiseux series, obtained by adjoining roots of t. [35]