Euclidean domain

In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain (also called a Euclidean ring) is a commutative ring that can be endowed with a Euclidean function (explained below) which allows a suitable generalization of the Euclidean division of the integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.

It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same "structural properties" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and Bézout's identity. In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra.

So, given an integral domain R, it is often very useful to know that R has a Euclidean function: in particular, this implies that R is a PID. However, if there is no "obvious" Euclidean function, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain.

Euclidean domains appear in the following chain of class inclusions:

commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields

Definition

Let R be an integral domain. A Euclidean function on R is a function $f$ from $R \setminus \{0\}$ to the non-negative integers satisfying the following fundamental division-with-remainder property:

(EF1) If a and b are in R and b is nonzero, then there are q and r in R such that a = bq + r and either r = 0 or f(r) < f(b).

A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. 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.

Most algebra texts require a Euclidean function to have the following additional property:

(EF2) For all nonzero a and b in R, f(a) ≤ f(ab).

However, one can show that (EF2) is superfluous in the following sense: any domain R which can be endowed with a function g satisfying (EF1) can also be endowed with a function f satisfying (EF1) and (EF2): indeed, for $\scriptstyle a \in R \setminus \{0\}$ one can define f(a) as follows^[1]

f(a) = \min_{x \in R \setminus \{0\}} g(xa)

In words, one may define f(a) to be the minimum value attained by g on the set of all non-zero elements of the principal ideal generated by a.

A multiplicative Euclidean function is one such that f(ab)=f(a)f(b) and f(a) is never zero. It follows that f(1)=1 and in fact f(a)=1 if and only if a is a unit.

Notes on the definition

Many authors use other terms such as "degree function", "valuation function", "gauge function" or "norm function", in place of "Euclidean function". Some authors also require the domain of the Euclidean function to be the entire ring R;^[2] however this does not essentially affect the definition, since (EF1) does not involve the value of f(0). The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set; this weakening does not affect the most important implications of the Euclidean property.

The property (EF1) can be restated as follows: for any principal ideal I of R with nonzero generator b, all nonzero classes of the quotient ring R/I have a representative r with f(r) < f(b). Since the possible values of f are well-ordered, this property can be established by proving f(r) < f(b) for any r (not in I) with minimal value of f(r) in its class. Note that for a Euclidean function that is so established there need not exist an effective method to determine q and r in (EF1).

Examples

Examples of Euclidean domains include:

Any field. Define f(x) = 1 for all nonzero x.
Z, the ring of integers. Define f(n) = |n|, the absolute value of n.^[3]
Z[i], the ring of Gaussian integers. Define f(a + bi) = a² + b², the squared norm of the Gaussian integer a + bi.
Z[ω] (where ω is a primitive (non-real) cube root of unity), the ring of Eisenstein integers. Define f(a + bω) = a² − ab + b², the norm of the Eisenstein integer a + bω.
K[X], the ring of polynomials over a field K. For each nonzero polynomial P, define f(P) to be the degree of P.^[4]
K''X'', the ring of formal power series over the field K. For each nonzero power series P, define f(P) as the degree of the smallest power of X occurring in P. In particular, for two nonzero power series P and Q, f(P)≤f(Q) iff P divides Q.
Any discrete valuation ring. Define f(x) to be the highest power of the maximal ideal M containing x (equivalently, to the power of the generator of the maximal ideal that x is associated to). The previous case K''X'' is a special case of this.
A Dedekind domain with finitely many nonzero prime ideals P₁, ..., P_n. Define $\scriptstyle f(x) = \sum_{i=1}^n v_i(x)$ , where $\scriptstyle v_i$ is the discrete valuation corresponding to the ideal P_i. (Samuel 1971)

Example of domains that are not Euclidean domains include

Every domain that is not a principal ideal domain, such as the ring of polynomials in at least two indeterminates over a field, or the ring of univariate polynomials with integer coefficients
The ring of integers of $\mathbb{Q}(\sqrt{-19}),$ consisting of the numbers $\frac{a+b\sqrt{-19}}{2}$ such that $a$ and $b$ are integers, that are either both even or both odd. It is a principal ideal domain that is not Euclidean.
The ring $\mathbb{R}[X,Y]/(X^2+Y^2+1)$ is also a principal ideal domain that is not Euclidean.

Properties

Let R be a domain and f a Euclidean function on R. Then:

R is a principal ideal domain (PID). In fact, if I is a nonzero ideal of R then any element a of I\{0} with minimal value (on that set) of f(a) is a generator of I.^[5] As a consequence R is also a unique factorization domain and a Noetherian ring. With respect to general principal ideal domains, the existence of factorizations (i.e., that R is an atomic domain) is particularly easy to prove in Euclidean domains: choosing a Euclidean function f satisfying (EF2), x cannot have any decomposition into more than f(x) nonunit factors, so starting with x and repeatedly decomposing reducible factors is bound to produce a factorization into irreducible elements.
Any element of R at which f takes its globally minimal value is invertible in R. If an f satisfying (EF2) is chosen, then the converse also holds, and f takes its minimal value exactly at the invertible elements of R.
If the Euclidean property is algorithmic, i.e., if there is a division algorithm that for given a and nonzero b produces a quotient q and remainder r with a = bq + r and either r = 0 or f(r) < f(b), then an extended Euclidean algorithm can be defined in terms of this division operation.^[6]
If a Euclidean domain is not a field then it has an element a with the following property: any element x not divisible by a can be written as x=ay+u for some unit u and some element y. This follows by taking a to be a non-unit with f(a) as small as possible. This strange property can be used to show that some principal ideal domains are not Euclidean domains, as not all PIDs have this property. For example, for d = −19, −43, −67, −163, the ring of integers of $\mathbb Q(\sqrt{d})$ is a PID which is not Euclidean, but the cases d = −1, −2, −3, −7, −11 are Euclidean.^[7]

However, in many finite extensions of Q with trivial class group, the ring of integers is Euclidean (not necessarily with respect to the absolute value of the field norm; see below). Assuming the extended Riemann hypothesis, if K is a finite extension of Q and the ring of integers of K is a PID with an infinite number of units, then the ring of integers is Euclidean.^[8] In particular this applies to the case of totally real quadratic number fields with trivial class group. In addition (and without assuming ERH), if the field K is a Galois extension of Q, has trivial class group and unit rank strictly greater than three, then the ring of integers is Euclidean.^[9] An immediate corollary of this is that if the number field is Galois over Q, its class group is trivial and the extension has degree greater than 8 then the ring of integers is necessarily Euclidean.

Norm-Euclidean fields

Algebraic number fields K come with a canonical norm function on them: the absolute value of the field norm N that takes an algebraic element α to the product of all the conjugates of α. This norm maps the ring of integers of a number field K, say O_K, to the nonnegative rational integers, so it is a candidate to be a Euclidean norm on this ring. If this norm satisfies the axioms of a Euclidean function then the number field K is called norm-Euclidean or simply Euclidean.^[10]^[11] Strictly speaking it is the ring of integers that is Euclidean since fields are trivially Euclidean domains, but the terminology is standard.

If a field is not norm-Euclidean then that does not mean the ring of integers is not Euclidean, just that the field norm does not satisfy the axioms of a Euclidean function. In fact, the rings of integers of number fields may be divided in several classes:

Those that are not principal and therefore not Euclidean, such as the integers of $\mathbb{Q}(\sqrt{-5})$
Those that are principal and not Euclidean, such as the integers of $\mathbb{Q}(\sqrt{-19})$
Those that are Euclidean and not norm-Euclidean, such as the integers of $\mathbb{Q}(\sqrt{69})$ ^[12]
Those that are norm-Euclidean, such as Gaussian integers (integers of $\mathbb{Q}(\sqrt{-1})$ )

The norm-Euclidean quadratic fields have been fully classified, they are $\mathbb{Q}(\sqrt{d})$ where d takes the values

−11, −7, −3, −2, −1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 (sequence A048981 in OEIS).^[13]

Every Euclidean imaginary quadratic field is norm-Euclidean and is one of the five first fields in the preceding list.

Notes

↑ Rogers, Kenneth (1971), "The Axioms for Euclidean Domains", American Mathematical Monthly (Mathematical Association of America) 78 (10): 1127–1128, doi:10.2307/2316324, JSTOR 2316324, Zbl 0227.13007
↑ E.g., Dummit and Foote, Abstract Algebra, 3rd edition, p. 270
↑ Fraleigh & Katz (1967), p. 377, Example 1
↑ Fraleigh & Katz (1967), p. 377, Example 2
↑ Fraleigh & Katz (1967), p. 377, Theorem 7.4
↑ Fraleigh & Katz (1967), p. 380, Theorem 7.7
↑ Motzkin, Theodore (1949), "The Euclidean algorithm", Bulletin of the American Mathematical Society 55 (12): 1142–1146, doi:10.1090/S0002-9904-1949-09344-8, Zbl 0035.30302
↑ Weinberger, Peter J. (1973), "On Euclidean rings of algebraic integers", Proceedings of Symposia in Pure Mathematics (AMS) 24: 321–332
↑ Harper, Malcolm; Murty, M. Ram (2004), "Euclidean rings of algebraic integers" (PDF), Canadian Journal of Mathematics 56 (1): 71–76, doi:10.4153/CJM-2004-004-5
↑ Ribenboim, Paulo (1972). Algebraic Numbers. Wiley-Interscience. ISBN 0-471-71804-1.
↑ Hardy, G. H. and Wright, E. M. (1975). An Introduction to The Theory of Numbers. Oxford.
↑ Clark, David A. (1994). "A quadratic field which is Euclidean but not norm-Euclidean". Manuscripta Mathematica 83 (3-4): 327–330. doi:10.1007/BF02567617. Zbl 0817.11047.
↑ LeVeque, William J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. pp. II:57,81. ISBN 978-0-486-42539-9. Zbl 1009.11001.

References

John B. Fraleigh, Victor J. Katz. A first course in abstract algebra. Addison-Wesley Publishing Company. 5 ed., 1967. ISBN 0-201-53467-3
Pierre Samuel, "About Euclidean rings", Journal of Algebra 19 (1971) 282-301.

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