At the same time, Kummer developed powerful new methods to prove Fermat's Last Theorem at least for a large class of prime exponents using what we now recognize as the fact that the ring is a Dedekind domain.

In fact Kummer worked not with ideals but with " ideal numbers ", and the modern definition of an ideal was given by Dedekind.

Another illustration of the delicate / robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local property -- a Noetherian domain is Dedekind iff for every maximal ideal of the localization is a Dedekind ring.

In fact, this is the definition of a Dedekind domain used in Bourbaki's " Commutative algebra ".

However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal ( except R ) is a product of prime ideals.

In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain.

In this domain the simple fact of coexistence in the same local, national, and world community is enough to guarantee that we cannot refrain from having some effect, large or small, upon Gentile-Jewish relations.

The entire functions on the complex plane form an integral domain ( in fact a Prüfer domain ).

With the exception of the possible CCA cryptographic attack described in the limitations section below, the attacker can only know the fact that a connection is taking place between the two parties, already known to him, the domain name and IP addresses.

In fact, f coincides with its Taylor series at a in any disk centered at that point and lying within the domain of the function.

In fact the activities banned on the Sabbath are not " work " in the usual sense: They include such actions as lighting a fire, writing, using money and carrying in the public domain.

However, many others are either lost or in the public domain ( in fact, per US law, all films released before 1923 are in the public domain ).

The common contemporary Latin legal term used in documents of the Holy Roman Empire was for a long time regnum (" rule, domain, empire ", such as in Regnum Francorum for the Frankish Kingdom ) before imperium was in fact adopted, the latter first attested in 1157, whereas the parallel use of regnum never fell out of use during the Middle Ages.

The French prefix sur means over or above and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain.

The fact that the density is positive definite and convected according to this continuity equation, implies that we may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law.

If in addition all of the elements in the codomain are in fact mapped to by some element of the domain, then the function is said to be bijective ( see figures ).

Note that this application is not a contradiction to the fact that conformal mappings preserve angles, they do so only for points in the interior of their domain, and not at the boundary.

Cao Pi's strategist Liu Ye suggested that Cao Pi decline — and in fact attack Sun Quan on a second front, effectively partitioning Sun's domain with Shu, and then eventually seek to destroy Shu as well.

If R = K is a field, then K (( X )) is in fact a field, which may alternatively be obtained as the field of fractions of the integral domain K < nowiki ></ nowiki > X < nowiki ></ nowiki >.

To bring the numerical values of parameters μ, σ into the domain where strong inequality holds true one could use the fact that if X is log-normally distributed then X < sup > m </ sup > is also log-normally distributed with parameters μm, σm.

The frequency spectrum of the modulated color signal overlaps that of the baseband signal, and separation relies on the fact that frequency components of the baseband signal tend to be near harmonics of the horizontal scanning rate, while the color carrier is selected to be an odd multiple of half the horizontal scanning rate ; this produces a modulated color signal that consists mainly of harmonic frequencies that fall between the harmonics in the baseband luma signal, rather than both being in separate continuous frequency bands alongside each other in the frequency domain.

In fact, one important aspect of their very religion is the annihilation of men ''.

In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.

The fact is due mainly to international wars, both hot and cold.

While the pattern is uneven, some having gained more than others, nationalism has in fact served the Western peoples well.

In point of fact, this is a beige box with a bright red door, about one and a half feet square and hung from the wall about six feet from the door to Wisman's right.

The fact is that the Southern Confederacy differed from the earlier one almost as much as the Federal Constitution did.

To my knowledge, Lincoln remains the only Head of State and Commander-in-Chief who, while fighting a fearful war whose issue was in doubt, proved man enough to say this publicly -- to give his foe the benefit of the fact that in all human truth there is some error, and in all our error, some truth.

In fact the accumulation of the hardware of destruction is day by day increasing our fear of each other.

The new fact the initiates of this cult have to learn is that they must move toward simplicity.

The magic circle is, in fact, a symbol of and preparation for the metaphysical orgasm ''.

Operating as a one man police force in fact if not in name, he is at once more independent and more dedicated than the police themselves.

`` I may possibly be a greater risk than is the normal person of my age '', the President had said on February 29th of the election year, ignoring the fact that no one of his age had ever lived out another term.

In the incessant struggle with recalcitrant political fact he learns to focus the essence of a problem in the significant detail, and to articulate the distinctions which clarify the detail as significant, with what is sometimes astounding rapidity.

we accord it its place there, and in Lawrence's treatment we are given the innocent fantasy of a child, in fact, the form in which oedipal love is expressed in childhood.

There is probably some significance in the fact that two of the best incest stories I have encountered in recent years are burlesques of the incest myth.

How much they esteemed him is shown by the fact that their underground committee selected him as one of the few who would be helped to escape.

That is not to deny that he has been aware of traditions, of course, that he is steeped in them, in fact, or that he has dealt with them, in his books.

There is evidence to suggest, in fact, that many authors of the humorous sketches were prompted to write them -- or to make them as indelicate as they are -- by way of protesting against the artificial refinements which had come to dominate the polite letters of the South.

It seems quite obvious that all the really difficult tasks of human beings arise from the fact that man is not one, but many.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

Perhaps the mere fact that by plucking on the nerves nature can awaken in the most ordinary of us, temporarily anyway, the sleeping poet, and in poets can discover their immortality, is the most remarkable of all the remarkable phenomena to which we can attest??