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An extension is algebraic if and only if its transcendence degree is 0; the empty set serves as a transcendence basis here.; The field of rational functions in n variables K(x 1,...,x n) (i.e. the field of fractions of the polynomial ring K[x 1,...,x n]) is a purely transcendental extension with transcendence degree n over K; we can for example take {x 1,...,x n} as a transcendence base.
For example, the infinite sequence (,, … ) {\displaystyle (1,2,\ldots )} of the natural numbers increases infinitively and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has + ∞ {\displaystyle +\infty } as its least upper bound and as its limit (an actual infinity).
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The extension of life has been a desire of humanity and a mainstay motif in the history of scientific pursuits and ideas throughout history, from the Sumerian Epic of Gilgamesh and the Egyptian Smith medical papyrus, all the way through the Taoists, Ayurveda practitioners, alchemists, hygienists such as Luigi Cornaro, Johann Cohausen and ...
Extension also plays an important part in the philosophy of Baruch Spinoza, who says that substance (that which has extension) can be limited only by substance of the same sort, i.e. matter cannot be limited by ideas and vice versa. From this principle, he determines that substance is infinite.
It defines a realm like the one that contains the observable universe as only one of many inflationary zones. [1] [2]Astrophysicist Jean-Luc Lehners, of the Princeton Center for Theoretical Science, has argued that an inflationary universe does produce pockets.
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
The field extension C(T)/C, where C(T) is the field of rational functions over C, has infinite degree (indeed it is a purely transcendental extension). This can be seen by observing that the elements 1, T, T 2, etc., are linearly independent over C. The field extension C(T 2) also has infinite degree over C.