The primary water management objective is a comprehensive use of water empirycznych w przestrzeni Euklidesa, o wymiarach nie większych niż 3 [18]. the competition announced by the Society for Elementary Books in It C z e c h, J.: , Euklidesa Początków Geometryi ksiąg ośmioro, to jest. Primary 11 R 04; Secondary 11 H Key words and phrases. 6, – [ ] S. Lubelsky, Algorytm Euklidesa, Wiadom Mat. 42 (), 5–67 [] M.L.

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Documents Flashcards Grammar checker. Math Algebra The Euclidean algorithm elemenyt algebraic number fields advertisement. This article, which is an update of a version published in Expo.

We have also tried to put some emphasis on the open problems in this field. February 14, Mathematics Subject Classification. Primary 11 R 04; Secondary 11 H Key words and phrases. Euclidean algorithm, geometry of numbers. Definitions and General Properties 2. Euclidean Ideal Classes 3.

The Norm as a Euclidean Function 3. Lower Bounds for M K 3.

Euclidean Minima for k-stage Algorithms 4. Quadratic Number Fields 4. Complex Quadratic Number Fields 4. Real Quadratic Number Fields 5. Cubic Number Fields 5. Complex Cubic Number Fields 5.

Totally Real Cubic Number Fields 6. Quartic Number Fields leementy. Totally Complex Quartic Fields 6. Quartic Fields with Unit Rank 2 6. Totally Real Quartic Fields 7. Quintic Number Fields 8. How close Euclid came to understand the unique factorization property of the integers is open to debate: During the middle ages, Arabic and later European mathematicians studied the prime factors of a given number in connection with the problem of amicable numbers, and realized that the list of all factors of a number n can be produced from its prime factorization; the first clear statement of unique factorization, however, is due to Gauss euk,idesa The first mathematician who emphasized that the existence of a Eukliresa algorithm implied unique factorization was Dirichlet, and he did that as late as !

Definitions and General Properties An integral domain R is called Euclidean with respect to a given function f: There are equivalent definitions of Euclidean rings and functions, most of which are studied in []. For example, a function f: Variants of Euclidean functions euklidess been studied by Picavet [], Lenstra [], and Hiblot [97]. Obviously R is resp. An easy exercise shows Proposition 2. Then Z is S-Euclidean with respect to the usual absolute value.

Generalizations to k-ary algorithms were studied by Sorenson []. The last example shows that S-Euclidean rings are not necessarily Euclidean. I do not know whether Euclidean rings are S-Euclidean for suitably chosen sets S. It is easily seen that fmin is in fact a Euclidean function on R. The following observation is due to Motzkin []: InCooke [49] introduced the following more general concept: If we can replace 2. We also can introduce k-Euclidean minima in an obvious elemetny.

Euclidean ideal classes have been investigated by van der Linden [, ]. Non-trivial Euclidean ideal classes seem to occur very rarely: Schulze [] defined Euclidean systems; they generalize Euclidean ideal classes, and the simplest Euclidean systems correspond to the Dedekind-Hasse-test cf.

R is a principal ideal ring if and only if there is a function f: A different notion of a Euclidean system was introduced by Treatman in his thesis []. If the absolute value of the norm is a Euclidean function, OK or, by abuse of language, K is called norm-Euclidean. The Euclidean minimum elemeny K with ruklidesa to the norm is called norm-Euclidean minimum and will be denoted by M Euklidssa.

The first example of a norm-Euclidean ring OS was apparently given by Wedderburn1 []. The following theorem of Weinberger [] whose proof builds on previous work by Hooley suggested strongly the existence of number euklidesx that are Euclidean with respect to functions different from the norm GRH denotes a certain set of generalized Riemann hypotheses: Cooke [49, 50] shows unconditionally Proposition 3.

There are number fields with an infinite sequence of strictly decreasing Euclidean minima, and fields whose second minimum is not isolated.

Similar results not even a conjecture for fields with mixed signature are not elementg except for a theorem of Swinnerton-Dyer [] concerning complex cubic fields see Sect.

Lower Bounds for M K. There are several methods euklidssa getting bounds on M Kand in particular for showing that a given number field is not normEuclidean. The simplest criterion uses totally ramified primes: This was done as follows: Unfortunately, this method does not seem to work for other quadratic number fields; there are, duklidesa, numerous examples in degree 3 cf.

See Mandavid [] for a detailed exposition. Euclidean Minima for k-stage Algorithms. Assume that GRH holds. George [81] for quadratic fields, and H. Cohen [44] in general. Complex Quadratic Number Fields.

In order to prove Prop. Proofs for this fact have later been given by Birkhoff [13] and Schatunowsky []. Concerning k-stage Euclidean rings, we have the result of P. Wilson [], Campoli [21], Feyzioglu [75]. The results of Prop. Real Quadratic Number Fields. The real quadratic number fields that are norm-Euclidean are known: The if-part of Thm. The following table shows the evolution of the proof: In a letter to Perron see []Schur shows that Q 47 is not norm-Euclidean.

We know the following bounds for Euclidean minima of quadratic fields: Davenport [53, 54, 55] for elsmenty examples: On the other hand we know cf.

## Euclidean Algorithm

It seems likely that the xk generate C2which would imply that M2 is attained. The Euclidean and inhomogeneous minima Mi K of real quadratic fields K may or may not have the following properties: Mi K is attained;: Ci K is finite;: Mi K is isolated;: Moreover, E1 is true, and we conjecture that I1 always holds.

The following combinations are known to occur: This leaves, of course, a lot of questions unanswered: The following observation concerning Euclidean windows can be proved easily using ideals of small norm: So far it has not been possible to prove that the Euclidean window of p is non-empty using the method of Clark that succeeds for D 69 ; even a modification of this idea due to R.

Complex Cubic Number Fields.

### Euclidean Algorithm — from Wolfram MathWorld

See the tables at the end of this survey for known results on Euclidean minima of cubic fields. In the tables below, let E denote the number of fields in a given interval which are norm-Euclidean; the number of those which are not norm-Euclidean will be denoted by N. As in the quadratic case it is possible to compute the Euclidean minima of an infinite sequence of fields: This sequence incidentally shows that the upper bound in Prop. Similar results for sufficiently large a are known for other families of cubic number fields cf.

Computer calculations have led to the following Conjecture.

## The Euclidean algorithm in algebraic number fields

Totally Real Cubic Number Fields. Clark [40] independently has shown some fields to be norm-Euclidean.

Heilbronn [94] proved that the number of norm-Euclidean cyclic cubic fields is finite, but could give no bound for the discriminants of such fields. Non-cyclic Totally Real Fields.