*PM : proof that an Euclidean domain is a PID, id = 3015-- WP guess : proof that an Euclidean domain is a PID-- Status:
32.
;"'Euclidean domain "': A " Euclidean domain " is an integral domain in which a degree function is defined so that " division with remainder " can be carried out.
33.
;"'Euclidean domain "': A " Euclidean domain " is an integral domain in which a degree function is defined so that " division with remainder " can be carried out.
34.
In some circumstances these coincide : the special linear group over a field or a Euclidean domain is generated by transvections, and the " stable " special linear group over a Dedekind domain is generated by transvections.
35.
If the value of " x " can always be taken as 1 then " g " will in fact be a Euclidean function and " R " will therefore be a Euclidean domain.
36.
It is important to note that a particular Euclidean function " f " is " not " part of the structure of a Euclidean domain : in general, a Euclidean domain will admit many different Euclidean functions.
37.
It is important to note that a particular Euclidean function " f " is " not " part of the structure of a Euclidean domain : in general, a Euclidean domain will admit many different Euclidean functions.
38.
As in the Euclidean domain, the " size " of the remainder ? 0 must be strictly smaller than ?, and there must be only a finite number of possible sizes for ? 0, so that the algorithm is guaranteed to terminate.
39.
If these variables are specialized to be all value q, then this is called the q-matrix ( over the Euclidean domain \ mathbb { Q } [ q ] ) for the arrangement and much information is contained in its Smith normal form.
40.
Using the facts that the Gaussian integers are a Euclidean domain and that for a Gaussian integer p | p | ^ 2 is always a square it is possible to show that a Pythagorean triples correspond to the square of a prime Gaussian integer if the hypotenuse is prime.
