Help


[permalink] [id link]
+
Page "Algebraic closure" ¶ 10
from Wikipedia
Edit
Promote Demote Fragment Fix

Some Related Sentences

For and finite
For finite sets X, the axiom of choice follows from the other axioms of set theory.
For example, the number of solutions of an equation over a finite field reflects the topological nature of its solutions over the complex numbers.
For example, intervals, where takes all integer values in Z, cover R but there is no finite subcover.
For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.
For a finite group, the derived series terminates in a perfect group, which may or may not be trivial.
For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite ordinal numbers via transfinite recursion, thereby obtaining the transfinite derived series, which eventually terminates at the perfect core of the group.
For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.
For continuous dynamical systems, the map τ is understood to be a finite time evolution map and the construction is more complicated.
* For every prime number p and positive integer n, there exists a finite field with p < sup > n </ sup > elements.
For example, Graham's number, though finite, is unimaginably larger than other well-known large numbers such as a googol, googolplex, and even larger than Skewes ' number and Moser's number.
For some finite n-valued logics, there is an analogous law called the law of excluded n + 1th.
For every finite dimensional matrix Lie algebra, there is a linear group ( matrix Lie group ) with this algebra as its Lie algebra.
Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values.
For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.
For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population.
For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite.
For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits.
For a nonzero vector of finite norm in, one can assume that is nonzero, or to fix ideas.
For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup.
For any luminosity from a given distance L ( r ) N ( r ) proportional to r < sup > a </ sup >, is infinite for a ≥ − 1 but finite for a < − 1.
For point masses the gravitational energy decreases without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative ( since it decreases from zero ) for smaller finite distances.
For a set of polynomial equations in several unknowns, there are algorithms to decide if they have a finite number of complex solutions.

For and field
For example, there are persons who are in physical science, in the field of mineralogy, trained in crystallography, who use only X-rays, applying only the powder technique of X-ray diffraction, to clay minerals only, and who have spent the last fifteen years concentrating on the montmorillonites ; ;
For it is their catastrophic concept of the Anglo-Saxon invasions rather than Kemble's gradualist approach which dominates the field.
For example, the field extension R / Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C / R and Q (√ 2 )/ Q are algebraic, where C is the field of complex numbers.
For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers.
For example, if K is a field of characteristic p and if X is transcendental over K, is a non-separable algebraic field extension.
For field ion microscopy a gas is introduced, known as the imaging gas, into the chamber at very low pressures.
For these tip models, solutions to the field may be approximated or obtained analytically.
For one thing they had too good an opinion of their own ability ... Another point was their faulty field dispositions, and in addition there was rampant indiscipline and inexperience displayed ...
For example, it also predicts the variation of the particle density in a gravitational field with height, if.
For instance, Cooper pairs have been observed in ultracold gases of Fermions where a homogeneous magnetic field has been tuned to their Feshbach resonance.
For example, the CIA World Factbook uses the word in its " Country name " field to refer to " a wide variety of dependencies, areas of special sovereignty, uninhabited islands, and other entities in addition to the traditional countries or independent states ".< sup id = Note_1 >
For example, a 1039 bit integer was factored with the special number field sieve using 400 computers over 11 months.
For example, if an electric field is placed across a solution of Na < sup >+</ sup > and Cl < sup >−</ sup > ( and conditions are right ) the sodium ions move towards the negative electrode ( cathode ), while the chloride ions move towards the positive electrode ( anode ).
* For cavitation in the engineering field, visit
* For hydrodynamic cavitation in the ethanol field, visit and Ethanol Producer Magazine: " Tiny Bubbles to Make You Happy "
For example, the following line from the Aeneid ( VIII. 596 ) describes the movement of rushing horses and how " a hoof shakes the crumbling field with a galloping sound ":

For and prime
For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N / ln N. Therefore the proportion of prime integers is roughly 1 / ln N, which tends to 0.
For example, the set with elements 2, 3, and 5 is equal to the set of all prime numbers less than 6.
For example, the Riemann hypothesis is a conjecture from number theory that ( amongst other things ) makes predictions about the distribution of prime numbers.
For 50 years, Coronation Street has remained at the centre of ITV's prime time schedule.
For example, primality testing is the problem of determining whether a given number is prime or not.
For a prime p, the p-adic numbers arise by completing the rational numbers with respect to a different metric.
For each of the prime fields, one elliptic curve is recommended.
For example, neither 6 nor 35 is a prime number, since they both have two prime factors: 6 = 2 × 3 and 35 = 5 × 7.
The GCD of two numbers a and b is the product of the prime factors shared by the two numbers, where a same prime factor can be used multiple times, but only as long as the product of these factors divides both a and b. For example, since 1386 can be factored into 2 × 3 × 3 × 7 × 11, and 3213 can be factored into 3 × 3 × 3 × 7 × 17, the greatest common divisor of 1386 and 3213 equals 63 = 3 × 3 × 7, the product of their shared prime factors.
For example, 9 and 28 are relatively prime.
For example, Campbell states that, " There have been systems of religion where the mother is the prime parent, the source ... We talk of Mother Earth.
For prime minister, he selected Odilon Barrot, an unobjectionable middle-road parliamentarian, who had led the " loyal opposition " under Louis-Philippe.
For most of Nehru's term as prime minister, India would continue to face serious food shortages despite progress and increases in agricultural production.
For instance, Goldbach's conjecture is the assertion that every even number ( greater than 2 ) is the sum of two prime numbers.
For many people choosing an office suite, prime considerations include how valuable having learned that office suite will prove to potential employers, and how well the software interoperates with other users.
For example, 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2 and 3 in addition to 1 and 6.

0.154 seconds.