has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

By setting K = ker ( f ) we immediately get the first isomorphism theorem.

* By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.

By " satisfactory " one would mean at least the equivalent of Plancherel theorem.

By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization.

By the Glivenko – Cantelli theorem, if the sample comes from distribution F ( x ), then D < sub > n </ sub > converges to 0 almost surely.

By this theorem, once a star's chemical composition and its position on the main sequence is known, so too is the star's mass and radius.

By Gödel's incompleteness theorem, Peano arithmetic is incomplete and its consistency is not internally provable.

By Rice's theorem, the 1-halting problem is undecidable.

By the corollary to the recursion theorem, there is an index such that returns.

By Liouville's theorem, Hamiltonian flows preserve the volume form on the phase space.

By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation.

By the first isomorphism theorem, the image of A under ƒ is a substructure of B isomorphic to the quotient of A by this congruence.

By the central limit theorem, this distribution approaches the normal distribution as n increases.

By analogy to the prior and posterior probability terms in Bayes ' theorem, Bayes ' rule can be seen as Bayes ' theorem in odds form.

By the extreme value theorem, a continuous function on a closed interval must attain its minimum and maximum values at least once.

By the rank-nullity theorem, a system of n vectors in k dimensions ( where all dimensions are necessary ) satisfies a ( p = n − k )- dimensional space of relations.

By Tychonoff's theorem we have that is compact since is, so the closure of in is a compactification of.

By Stone's representation theorem every Boolean ring is isomorphic to a field of sets ( treated as a ring with these operations ).

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

By Euler's rotation theorem, we may replace the vector with where is a 3x3 rotation matrix and is the position of the particle at some fixed point in time, say t = 0.

By the divergence theorem Gauss's law can alternatively be written in the differential form:

By the Gelfand theorem, a commutative C *- algebra is isomorphic to the C *- algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C *- algebra up to homeomorphism.

By Gelfand representation, every commutative C *- algebra A is of the form C < sub > 0 </ sub >( X ) for some locally compact Hausdorff X.

By winning the Chess World Cup 2009, Gelfand qualified for the World Chess Championship 2012 Candidates Tournament.

By knowing the coordinates of an object ( usually given in equatorial coordinates ), the telescope user can use the setting circle to align the telescope in the appropriate direction before looking through its eyepiece.

By the River Styx, he sees the souls of those not given a proper burial, forced to wait by the river until someone buries them.

By contrast, while defendants in most civil law systems can be compelled to give a statement, this statement is not subject to cross-examination by the prosecutor and not given under oath.

By convention, the bicyclic β-lactams are numbered starting with the position occupied by sulfur in the penams and cephems, regardless of which atom it is in a given class.

By the well ordering principle, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property ; therefore, there is a smallest positive integer satisfying the property " not definable in under eleven words ".

By contrast, in civil law jurisdictions ( the legal tradition that prevails in, or is combined with common law in, Europe and most non-Islamic, non-common law countries ), courts lack authority to act where there is no statute, and judicial precedent is given less interpretive weight ( which means that a judge deciding a given case has more freedom to interpret the text of a statute independently, and less predictably ), and scholarly literature is given more.

By the next day the Doge and the leading Franks were installed in the Great Palace, and the city was given over to pillage for three days.

By this time, portions of Ormsby County had been given over to neighboring counties, reducing the county to Carson City and a few outlying settlements.

By the time the government killed the project in 1842, they had given Babbage over ₤ 17, 000, more than double the cost of a warship, without receiving a working engine.

By construction, each point on the curve shows productive efficiency in maximizing output for given total inputs.

By mistake, the year of his birth was given as 1848, not the correct 1845.

By 27 June, the royal party had overtly given in, although the military began to arrive in large numbers around Paris and Versailles.

By comparing the genome compositions between genomes, scientists can better understand the evolutionary history of a given genome.

By this solution though, any given object always has all the properties throughout time, and the properties are merely temporally-specific.

By age 15, he had already given concerts worldwide.

By 1941, Germany had given up hope of evicting the Allies from the island and the constant air raids stopped.

By June 2003, the lower house of Parliament announced that President Akayev and two other " puppet " leaders of Kyrgyzstan, from the Soviet era, would be given lifetime immunity from prosecution, raising the prospect of Akayev finally stepping down.

By imperial proclamation at the wedding, Gustav was given the additional surname " Krupp ," which was to be inherited by primogeniture along with the company.

By the turn from the 10th to the 11th century, the smaller burghs in the Parseta area were given up.

By 1992 the problem had become so serious that Latvian forestry officials were given the right to carry firearms.

By November 1920 Theremin had given his first public concert with the instrument, now modified with a horizontal volume antenna replacing the earlier foot-operated volume control.