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Some Examples of Principal Ideal Domain Which Are Not Euclidean and Some Other Counterexamples

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Some Examples of Principal Ideal Domain Which Are Not Euclidean and Some Other Counterexamples Novi Sad J. Math. Vol. 38, No. 1, 2008, 137-154 SOME EXAMPLES OF PRINCIPAL IDEAL DOMAIN WHICH ARE NOT EUCLIDEAN AND SOME OTHER COUNTEREXAMPLES Veselin Peri¶c1, Mirjana Vukovi¶c2 Abstract. It is well known that every Euclidean ring is a principal ideal ring. It is also known for a very long time that the converse is not valid. Counterexamples exist under the£p rings¤ R of integral algebraic numbers in quadratic complex ¯elds Q ¡D , for D = 19; 43; 67, and 163. In conection with these counterexamples several results were published in an e®ort to make them somewhat more accessible. The aim of this note is to present and complete these results. AMS Mathematics Subject Classi¯cation (2000): 13F07, 13F10, 13F15, 11R04 Key words and phrases: Principal ideal domains, Euclidean domains, Unique factorization domains, Rings of algebraic integers in some quadra- tic ¯eld 0. Introduction It is well known that any Euclidean domain is a principal ideal domain, and that every principal ideal domain is a unique factorization domain. The main examples of Euclidean domains are the ring Z of integers and the polynomial ring K[x] in one variable x over a ¯eld K. It is known that the polynomial ring R[x] in one variable x over a unique factorization domain R is also a unique factorization domain. So the ring Z[x1; :::; xn] of all polynomi- als in n ¸ 1 variables over Z, and the polynomial ring K[x 1, ...,x n], in n ¸ 1 variables over K, are also unique factorization domains. But, the ring Z[x 1, ...,x n] for n ¸ 1, and the ring K[x1; :::; xn] for n ¸ 2, are not principal ideal domains. Therefore, we have simple examples of unique factorization do- mains, that are not principal ideal domains. It is easy to see that any Euclidean domain is a principal ideal domain. But, usually, to show that the converse is not valid, one gives no counterexamples, or one refers to [3]. Unfortunately, for such examples one cannot ¯nd all details which could make those examples easy to understand, in [3]. The counterexamples existp under the rings R of integral algebraic numbers in quadratic complex ¯elds Q[ ¡D], for D = 19, 43, 67, 163, ([6], 1967). Namely, 1Department of Mathematics, University of Montenegro, Podgorica, Montenegro, e-mail: [email protected] 2Department of Mathematics, University of Sarajevo, Sarajevo, Bosnia and Herzegovina, e-mail: [email protected] 138 V. Peri¶c,M. Vukovi¶c H. M. Stark in the mentioned article proved thatp under the rings R of integral algebraic numbers in complex quadratic ¯elds Q[ ¡D], exactly those for D = 1, 2, 3, 7, 11, 19, 43, 67, and 163, are principal ideal domains, and it is well known (see ([2], 1962, Th. 246, p. 213) that the ¯rst ¯ve of these rings are also Euclidean domains. Toward the end of the 20th century some articles ([8], 1973; [1], 1988) were published in which for one of the remaining four rings (for that with D=19) it was proved that it is a principal ideal, but not a Euclidean domain. In another article ([7], 1975), taking in account that all remaining rings R (for D = 19, 43, 67, and 163) are principal ideal domains, it was proved that none of those rings are Euclidean domains. The last proof is somewhat simpler than the corresponding proof in [8], because in this ¯rst proof some non-essential details were omitted. Both of these proofs are based on a theorem from [3]. The corresponding proof in [1] is not directly based on the cited theorem, but it is essentially not di®erent from the proof in [7]. The proofs in [8] and [1], that, for D = 19, the ring R is a principal ideal domain, di®er slightly, and are based on a theorem in [7], which is due to Dedekind and Hasse. So, in view of [8] and [1], respectively [7], we have one, respectively four, counterexamples of rings which are principal ideals, but not Euclidean domains. Those counterexamples are easier to understand and there are no other such counterexamples underp the rings R of integral algebraic numbers in the complex quadratic ¯elds Q [ ¡D]. Let us remark that underp rings R of integral algebraic numbers in the com- plex quadratic ¯elds Q[ ¡D] there are such rings that are not unique factor- ization domains (for instance, for D = 5). The aim of this article is to present all of the mentioned results from [8], [7], and [1].p For this purpose we ¯rst describe (Section 1) the complex quadratic ¯eld Q[ ¡D] and the ring R of integral algebraic numbers R in this ¯eld. Then we give some basic facts (Section 2) about Euclidean, "almost Euclidean", and principal ideal domains, including the two mentioned theorems from [3] which we will use in the last section (Section 3). The last section contains all mentioned and some other examples and counterexamples, from which those which are well known were only mentioned. 1. The ring R of all integralp algebraic numbers in the com- plex quadratic ¯eld Q[ ¡D] p First we are concerned with the complex quadratic ¯eld Q[ ¡D] and with the ring R of all algebraic integers in this ¯eld. Let Q be the ¯eld of rational numbers,p and D pbe a positive integer with- out square factors greater than 1. Then ¡D = i D is a root of the monic quadratic equation p p (X ¡ ¡D)(X ¡ ¡D) = 0 Some examples of principal ideal domains which are not Euclidean and ... 139 with the rational coe±cients: p p p p 1; ¡( ¡D + ¡D) = 0; ¡D ¢ ¡D = D; hence, an algebraic number of degree 2. Theorem 1.1. The set p p Q[ ¡D] = fa + b ¡D j a; b 2 Qg is a sub¯eld of thep ¯eld C of all complex numbers, actually the smalest sub¯eld containing Q and ¡D. Proof. For a = 0 and b = 1, we have p p p ¡D = 0 + 1 ¢ ¡D 2 Q[ ¡D]; and for a 2 Q and b = 0, p p p a = a + 0 ¢ ¡D 2 Q[ ¡D]; i:e: Q ⊆ Q[ ¡D]: p Obviously, Q[ ¡D] is closed under addition and conjugation. Since p p p ¡D ¢ ¡D = ¡D 2 Q ⊆ Q[ ¡D]; p p the set Q[ ¡D] is also closed under multiplication. Therefore, Q[ ¡D] is a commutative ring withp unity element 1. In particular,p Q[ ¡D] is a vector space over the ¯eld Q. This space has a basis f1; ¡Dg, hence the dimension p dimQ Q[ ¡D] = 2: An element p p ® = a + b ¡D 2 Q[ ¡D]nf0g has in C an inverse ®¡1 = (®®)¡1®: But, ®® = a2 + b2D 2 Q; ®® 6= 0; i.e. p (®®)¡1 2 Q; hence (®®)¡1 ¢ ® 2 Q[ ¡D]: p p This means ®¡1 2 Q[ ¡D], and so Q[ ¡D] is a ¯eld. 2 p De¯nition 1.1. The ¯eld Q [ ¡D] is called the complex quadratic ¯eld of algebraic numbers. 140 V. Peri¶c,M. Vukovi¶c p De¯nition 1.2. For an element ® 2 Q [ ¡D] we say that it is an algebraic integer if the monic quadratic equation (x ¡ ®) ¢ (x ¡ ®) = 0 has all coe±cients in Z: 1; ¡(® + ®); ®® 2 Z: That is true for p ® = a + b ¡D; (a; b 2 Z); since ® + ® = 2a 2 Z; ®® = a2 + b2D 2 Z: But, in the case that (¤) D ´ ¡1(mod 4); i:e: D = 4k ¡ 1 (k ¸ 1) an algebraic number p p ® = a + b ¡D 2 Q[ ¡D] can be an algebraic integral number also when a2 = Z and=or b2 = Z: In this case we must have 2a + 1 ® + ® = 2a 2 Z; i:e: a = 1 (a 2 Z); 2 1 and 1 ®® = a2 + b2D = a2 + a + + b24k ¡ b2 2 Z; 1 1 4 hence 1 1 a = a + ; b = b + ; (a ; b 2 Z): 1 2 1 2 1 1 Under the additional condition (¤), the algebraic number p ® = a + 1 + (b + 1 ) ¡D 1 2 1 2 p = (a1 ¡ b1) + (2b1 + 1)(1 + ¡D)=2; (a1; b1 2 Z); is also an integral algebraic number. In particular, for a1 = b1 = 0, 1 p p θ = (1 + ¡D) 2 Q[ ¡D] 2 is an algebraic integer. Thus, the following theorem is true: Some examples of principal ideal domains which are not Euclidean and ... 141 Theorem 1.2.p If the condition (¤) is ful¯lled, then for the set of all algebraic integers in Q[ ¡D] R = fa + bθ j a; b 2 Zg; holds, and otherwise, only p R = fa + b ¡D j a; b 2 Zg holds. In both cases R is an integral domain with unity element 1. In the ¯rst case p θ = (1 + ¡D)=2 is a root of the monic quadratic equation X2 ¡ X + k = 0; p where k is the rational integer in (¤), and ¡D is a root of the monic quadratic equation X2 + D = 0: De¯nition 1.3. The ring R in Theoremp 1.2. is called the ring of algebraic integers in the complex quadratic ¯eld Q[ ¡D]. This ring is closed under conjugation, since p p θ = k ¡ θ; respectively ¡D = ¡ ¡D: p In the ring R of algebraic integers in Q[ ¡D] we have the mapping ' : R ! Z;'(®) = ®®; (® 2 R) with the properties: i) '(®) ¸ 0;'(®) = 0 , ® = 0; ii) '(®¯) = '(®) ¢ '(¯); (®; ¯ 2 R). From ii), it follows ii') '(®¯) ¸ '(®); (®; ¯ 2 R; ¯ 6= 0). Using the mapping ', which has the ¯rst two properties of the Euclidean norm in the de¯nition of the Euclidean domain, we can determine the set U of all unities in R.
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