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A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term "infinite tuple" is occasionally used for "infinite sequences". Tuples are usually written by listing the elements within parentheses "( )" and separated by commas; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Other types of brackets are ...
The primitive element theorem was proved in its modern form by Ernst Steinitz, in an influential article on field theory in 1910, which also contains Steinitz's theorem; [6] Steinitz called the "classical" result Theorem of the primitive elements and his modern version Theorem of the intermediate fields.
Primitive element (lattice), an element in a lattice that is not a positive integer multiple of another element in the lattice; Primitive element (coalgebra), an element X on which the comultiplication Δ has the value Δ(X) = X⊗1 + 1⊗X; Primitive element (free group), an element of a free generating set
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing the first four positive integers (= {,,,}), one could say that "3 is an element of A", expressed notationally as .
For example, a Ducci sequence starting with the n-tuple (1, q, q 2, q 3) where q is the (irrational) positive root of the cubic = does not reach (0,0,0,0) in a finite number of steps, although in the limit it converges to (0,0,0,0).
In this case, a primitive element is also called a primitive root modulo q. For example, 2 is a primitive element of the field GF(3) and GF(5), but not of GF(7) since it generates the cyclic subgroup {2, 4, 1} of order 3; however, 3 is a primitive element of GF(7). The minimal polynomial of a primitive element is a primitive polynomial.
The function: () = [/, /], shown on the figure at the right, satisfies all Kakutani's conditions, and indeed it has many fixed points: any point on the 45° line (dotted line in red) which intersects the graph of the function (shaded in grey) is a fixed point, so in fact there is an infinity of fixed points in this particular case.
In mathematics, a finitary relation over a sequence of sets X 1, ..., X n is a subset of the Cartesian product X 1 × ... × X n; that is, it is a set of n-tuples (x 1, ..., x n), each being a sequence of elements x i in the corresponding X i. [1] [2] [3] Typically, the relation describes a possible connection between the elements of an n-tuple.