The Euclidean algorithm
Citations
139 citations
Cites background or methods from "The Euclidean algorithm"
...Motzkin [363] proved that the condition (21) is also necessary if R is to be Euclidean....
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...Motzkin [363] constructed the so-called minimal Euclidean algorithm....
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115 citations
Cites background from "The Euclidean algorithm"
...’s which are not Euclidean domains ([58]); expecially in Number Theory (see, for example, [48], [59] or [70]), one can find quadratic fields that are not Euclidean domains....
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110 citations
79 citations
Cites background or result from "The Euclidean algorithm"
...Motzkin Sets....
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...The following observation is due to Motzkin [147]:...
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...Motzkin Sets 4 2.4. k-stage Euclidean Rings 5 2.5....
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...The following observation is due to Motzkin [147]: Proposition 2.2....
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...For any integral domain R, define the Motzkin sets Ek, k ≥ 0, by E0 = {0}, E1 = {0} ∪R∗, the unit group of R and, generally, Ek = {0} ∪ {α ∈ R : each residue class mod α contains a β ∈ Ek−1}, E∞ = ⋃ k≥0 Ek The Motzkin sets of R = Z are easily computed: E0 = {0}, E1 = {0,±1}, E2 = {0,±1,±2,±3}, . . . , Ek = {0,±1 . . . ,±(2k − 1)}....
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59 citations
References
60 citations
39 citations
"The Euclidean algorithm" refers background or methods in this paper
...Further, such a tea determines, and is determined by, a transfinite sequence Sx, O ^ X ^ J U , of subsets of Q-0 with (1) Sx 'CSx+i, (2) S\QS\-i, but S\~C\Si, i<\, if X —1 does not exist, (3) empty S^....
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...It is easily seen that in every principal ideal ring the before mentioned norm 7* fulfils (1,1,2) (see, for example, [4, pp....
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...Let a tea (transfinite Euclidean algorithm) be an algorithm as before but where (1) we allow \b\ to take any ordinal numbers as values; (2) we do not require \a\ è |&| for 6 dividing a....
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...On the other hand, even the weakest condition with Z = l, which is (3, 3, 1), is not always fulfilled in principal ideal rings, as we have shown; and (3, 3, 1) is equivalent to (3, 2,1), (2 ,3, 1) is equivalent to (2, 2,1), (1, 3, l ) , and (1,2,1), and finally (2, 1, 1) to (1, 1, 1), while it remains open whether these three sets of conditions are really of different strength....
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15 citations
8 citations
"The Euclidean algorithm" refers background or methods in this paper
...Further, such a tea determines, and is determined by, a transfinite sequence Sx, O ^ X ^ J U , of subsets of Q-0 with (1) Sx 'CSx+i, (2) S\QS\-i, but S\~C\Si, i<\, if X —1 does not exist, (3) empty S^....
[...]
...It is easily seen that in every principal ideal ring the before mentioned norm 7* fulfils (1,1,2) (see, for example, [4, pp....
[...]
...Let a tea (transfinite Euclidean algorithm) be an algorithm as before but where (1) we allow \b\ to take any ordinal numbers as values; (2) we do not require \a\ è |&| for 6 dividing a....
[...]
...On the other hand, even the weakest condition with Z = l, which is (3, 3, 1), is not always fulfilled in principal ideal rings, as we have shown; and (3, 3, 1) is equivalent to (3, 2,1), (2 ,3, 1) is equivalent to (2, 2,1), (1, 3, l ) , and (1,2,1), and finally (2, 1, 1) to (1, 1, 1), while it remains open whether these three sets of conditions are really of different strength....
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