Separable

Definition: Separable

Separable

Adjective

1. Capable of being divided or dissociated; "often drugs and crime are not dissociable"; "the siamese twins were not considered seperable"; "a song...never conceived of as severable from the melody";.

Date "separable" was first used in popular English literature: sometime before 1609. (references)

Synonyms: Separable

Synonyms: dissociable (adj), severable (adj). (additional references)

In mathematics, an irreducible polynomial P(X) is separable if its roots in an algebraic closure are distinct - that is P(X) has distinct linear factors in some large enough field extension. This is the 'usual' case, since it turns out to be true if P is defined over a field K that is either (a) of characteristic 0, or (b) a finite field. This criterion is of technical importance in Galois theory.

If P is not assumed irreducible the concept is of lesser importance, since repeated roots may then just reflect that P isn't square-free . We can test for common factors of P(X) and the derivative P'(X) over any field, using the calculus formula: any repeated root will divide the highest common factor of P and P'. This leads to the quick conclusion that if P is irreducible and not separable, then P'(X)=0. This is only possible as a characteristic p phenomenon: we must have P(X) = Q(X^p) where the prime number p is the characteristic.

With this clue we can construct an example: P(X) = X^p - T with K the field of rational functions in the indeterminate T over the finite field with p elements. Here one can prove directly that P(X) is irreducible, and not separable. This is actually a typical example of why inseparability matters; in geometric terms P represents the mapping on the projective line over the finite field, taking co-ordinates to their p-th power. Such mappings are fundamental to the algebraic geometry of finite fields. Put another way, there are coverings in that setting that cannot be 'seen' by Galois theory.

If L is the field extension K(T^1/p) (the splitting field of P) then L/K is an example of a purely inseparable field extension. It is of degree p, but has no automorphism fixing K, other than the identity, because T^1/p is the unique root of P. This shows directly that Galois theory must here break down. A field such that there are no such extensions is called perfect. That finite fields are perfect follows a posteriori from their known structure.

One can show that the tensor product of fields of L with itself over K for this example has nilpotent elements that are non-zero. This is another manifestation of inseparability: that is, the tensor product operation on fields need not produce a ring that is a product of fields (so, not a commutative semisimple ring).

Separable space

(From Wikipedia, the free Encyclopedia)

In mathematics, a metric space (or topological space) X is separable if it has a countable subset Y such that members of Y approximate any x in X as closely as we like. Formally, a topological space is separable if and only if it has a subset that is both countable and dense. Separable spaces are therefore topological spaces with a certain limit on their size: an uncountable discrete space isn't separable.

Most of the spaces initially encountered are indeed separable: for example the real numbers with their standard metric have the rational numbers as a countable dense subset. Since the space of continuous functions on the interval [0,1] with the metric of uniform convergence has a dense subset of polynomials (see Weierstrass approximation theorem), and their coefficients can be approximated by rationals, that space is also separable. A Hilbert space is separable if and only if it has a countable orthonormal basis.

For technical reasons the foundations of general topology are written without the requirement of separability, or other 'axioms of countability'.

Separability is especially important in numerical analysis and constructive mathematics, since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs can be turned into algorithms for use in numerical analysis, and they are the only sorts of proofs acceptable in constructive analysis. A famous example of a theorem of this sort is the Hahn-Banach theorem.

Every second countable space is separable. As a partial converse, every separable metric space must be second countable. More generally, every separable uniform space whose uniformity has a countable basis must be second countable.

An example of a separable space that is not second countable is R_llt, the set of real numbers equipped with the lower limit topology. To avoid violating the previous paragraph, it follows that R_llt must not be metrisable -- it can't be made into a metric space. On the other hand, because R_llt is completely regular, it is uniformisable -- it can be made into a uniform space. But again, to avoid violating the previous paragraph, none of its uniformities could possibly have a countable basis.

Context	Synonyms within Context (source: adapted from Roget's Thesaurus).
Disjunction	Scissile, divisible, discerptible, partible, separable.
Source: adapted from Roget's Thesaurus.

Crosswords: Separable

English words defined with "separable": Bulbel ♦ Exemptitious ♦ family Lepiotaceae, Frondose ♦ Irresolvable ♦ Lepiotaceae ♦ Proximate principle ♦ Snap flask. (references)

Specialty definitions using "separable": flake graphite ♦ green book method ♦ intergrown ♦ lateral loss, lateral misalignment ♦ separable component part, separable cost, separable costs-remaining benefit method, Stripa process. (references)

Etymologies containing "separable": Exemptitious. (references)

Non-English Usage: "Separable" is also a word in the following language with English translations in parentheses.

Spanish (detachable, removable, separable).

Domain	Title
Books	Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations (Lecture Notes in Mathematics (Springer-Verlag), 1787.) (reference) Generic Requirements for Separable Electrical Connectors Used in Telecommunications Hardware Issue # 1 (reference) Separable Algebroids (Memoirs of the American Mathematical Society, 333) (reference) Separable extensions of quasi-Frobenius rings (reference) Separable Programming - Theory and Methods (Applied Optimization, Volume 53) (reference) (more book examples)
Source: compiled by the editor from various references; see credits.

Title	Author	Quote
Les Miserables	Hugo, Victor	To have Cosette, to possess Cosette, this to him was not separable from breathing.
Portrait of the Artist as a Young Man	Joyce, James	You apprehend it as complex, multiple, divisible, separable, made up of its parts, the result of its parts and their sum, harmonious.
Source: compiled by the editor from various references.

Speaker	Term	Phrase(s)
John Quincy Adams	1825-1829	Licentious blockades, irregularly enlisted or impressed sea men, and the property of honest commerce seized with violence, and even plundered under legal pretenses, are disorders never separable from the conflicts of war upon the ocean.
Source: compiled by the editor from various references.

"Separable" is generally used as an adjective (general or positive) -- approximately 100.00% of the time. "Separable" is used about 92 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted)

Parts of Speech	Percent	Usage per 100 Million Words	Rank in English
Adjective (general or positive)	100%	92	34,282

Expressions using "separable": separable component part ♦ separable cost ♦ separable element. Additional references.
Hypenated Usage
	Ending with "separable": non-separable.
Source: compiled by the editor from various references; see credits.

The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com.

Language		Translations for "separable"; alternative meanings/domain in parentheses.
Albanian		i ndashëm (detachable, divisible, fissile, fissionable, removable). (various references)

Arabic		‏ممكن فصله, ‏قابل للانفصال. (various references)

Bulgarian		отделим. (various references)

Chinese		可分开. (various references)

Czech		oddìlitelný (detachable, disjunctive). (various references)

Danish		sliddel (separable component part), udskiftningsdel (separable component part), omkostninger som kan udskilles (separable cost), modificeret kombinationsmetode (green book method, separable costs-remaining benefit method), aftagelig fælg (demountable rim, detachable rim, separable rim). (various references)

Dutch		scheidbaar componentdeel (separable component part), variabele kostprijscalculatie (green book method, separable costs-remaining benefit method), gedifferentieerde kosten (separable cost), demonteerbare velg (demountable rim, detachable rim, separable rim). (various references)

Farsi		مجزا (Apart, Discrete, Ditinct, Knockdown), قابل تفکیک , جداکردنی (Detachable), جداشدنی (Detachable, Dissoluble, Precipitant, Severable). (various references)

Finnish		eroavainen (different), eriävä (different, differing, diverging). (various references)

French		séparable. (various references)

German		trennbar (detachable, dividable, separably, severable). (various references)

Greek		διαχωριστόσ (fissile, severable), διαχωριστικόσ (segregative). (various references)

Hebrew		פרי�" (divisible), � תיק (detachable, removable). (various references)

Hungarian		szétválasztható, levehető (canopy, detachable), leválasztható, elválasztható. (various references)

Italian		separabile. (various references)

Japanese Kanji		可分 (divisible). (various references)

Japanese Katakana		かぶ�" (divisible, enquiring beforehand, excessive, generous, limited information, unmerited). (various references)

Korean		분리� 수 있�". (various references)

Manx		scarreydagh (diacritic, schismatic, sectarian, separatist). (various references)

Pig Latin		eparablesay

Portuguese		separável (detachable, dissocial, dissociate, fission, resoluble). (various references)

Romanian		separabil, detaşabil (detachable, removable). (various references)

Russian		отделимый (severable). (various references)

Serbo-Croatian		separabilan, odvojiv. (various references)

Spanish		separable (detachable, removable). (various references)

Swedish		skiljbar, isärtagbar (demountable, knockdown), avtagbar (removable). (various references)

Thai		แยกออกจากกันไ�"้. (various references)

Turkish		ayrılabilir (resolvable, sectional). (various references)

Ukranian		рознімний (built up, divided), віддільний. (various references)

Vietnamese		có thể tách ra (detachable), có thể phân ra (dissociable). (various references)
Source: compiled by the editor from various translation references.

Derivations
	Words beginning with "separable": separableness, separablenesses. (additional references)
	Words ending with "separable": inseparable. (additional references)
	Words containing "separable": inseparableness, inseparablenesses, inseparables. (additional references)
Misspellings
	"Separable" is suggested in spellcheckers for the following: Safarali, Searelle, separably, seperable. (additional references)
Source: compiled by the editor, based on several corpora (additional references).

Scrabble® Enable2K-Verified Anagrams
	Direct Anagrams: spareable.
	Words within the letters "a-a-b-e-e-l-p-r-s"
	-1 letter: erasable, parables, parsable, prebasal, reapable, sparable.
	-2 letters: arables, beleaps, earlaps, leapers, parable, pleaser, presale, relapse, repeals.
	-3 letters: abaser, abeles, arable, asleep, balers, beleap, blares, blears, bleeps, earlap, elapse, lapser, larees, leaper, leaser, lepers, paleae, parles, pearls, please, plebes, reales, rebels, repeal, repels, resale, reseal, salpae, sarape, sealer, serape.
	-4 letters: abase, abele, abler, ables, albas.
	Words containing the letters "a-a-b-e-e-l-p-r-s"
	+1 letter: spreadable.
	+2 letters: breastplate, inseparable, pallbearers, persuadable, pleasurable.
	+3 letters: breastplates, inseparables, perambulates.
	+4 letters: alphabetizers, separableness.
	+5 letters: comparableness, drapeabilities, pardonableness, separabilities.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.