Integral Domains Arising as Quotient Rings of Z[[x]] By James M. McDonough A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTERSIN SCIENCE (MATHEMATICS) at the CALIFORNIA STATE UNIVERSITY - CHANNEL ISLANDS 2011 © 2011 James M. McDonough ALL RIGHTS RESERVED Dedication To my friends and family, and especially my thesis advisor Dr. Jesse Elliott, who have supported me throughout this endeavor. iv Acknowledgements This was done with the help of my thesis advisor, Dr. Jesse Elliott, my commit- tee members, Dr. Ivona Grzegorzyck and Dr. Brian Sittinger, and technology advisor Dr. Jorge Garcia, all from California State University, Channel Islands. Camarillo, California August 17, 2011 v Abstract Using techniques of commutative algebra and p-adic analysis, we classify all integral domains arising as quotient rings of Z[[x]]. 1 Table of Contents 1 Introduction 3 2 Background on p-adic Numbers and Power Series Rings 6 2.1 p-adic Numbers . 6 2.2 Background on Ring Theory . 18 2.3 Power Series Rings . 20 3 Proofs and Examples 28 3.1 Proof of Main Theorem . 28 3.2 Examples . 38 3.3 Further Directions . 41 Bibliography 42 2 CHAPTER 1 Introduction While comparing the ring Z[x] of polynomials with integer coefficients with Z[[x]] its x-adic completion, the ring of formal power series with integer co- efficients, an interesting dilemma arises. While they do have some things in common, for instance, they are both Noetherian unique factorization do- mains (UFDs) of Krull dimension 2, one major difference between them has to do with known irreducibility criteria for their elements. For Z[x], there are known irreducibility criteria, such as Eisenstein’s criterion and Gauss’s Lemma. No such criteria are known for Z[[x]]. A 2008 paper by Daniel Bir- majer and Juan Gil (see [1]) gives examples of polynomials that are irreducible over Z[x], but reducible over Z[[x]], and vice versa. For example they show that 6 x x2 is irreducible over Z[x] but reducible over Z[[x]], and 2 7x 3x2 Å Å Å Å 3 CHAPTER 1. INTRODUCTION is irreducible over Z[[x]] but reducible over Z[x]. These examples lead to the following question. How do the quotient rings differ or compare between Z[x] and Z[[x]] when factoring out by one of these irreducible elements? The situation for Z[x] is well understood already. When factoring out Z[x] by an nonconstant irreducible polynomial, the resulting quotient ring is isomorphic to Z[®], where ® C is any root of the polyno- 2 2 h 1 p 23 i mial. For instance Z[x]/(6 x x ) Z ¡ Å ¡ . What can we say about Å Å Æ» 2 Z[[x]]/(f ), where f is irreducible? This answer to this question is not provided by any of the standard algebra texts. We will show, for example, that the ring Z[[x]]/(2 7x 3x2) is in fact isomorphic to the ring of 2-adic integers Z . Even Å Å 2 when a polynomial is irreducible over both Z[x] and Z[[x]], the quotient rings are generally not isomorphic. An example are the quotient rings Z[x]/(2 x2) ¡ and Z[[x]]/(2 x2), which are isomorphic to Z[p2] and Z [p2] respectively. ¡ 2 It is believed that a better understanding of the quotient rings of Z[[x]] may lead to clues about irreducibility criteria in Z[[x]]. Since Z[[x]] is a UFD, its prime elements and irreducible elements are the same; so when Z[[x]] is factored out by an ideal generated by an irreducible element the resulting quotient ring is an integral domain. Therefore, if one can classify the quo- tient rings that are integral domains, then the irreducibility characteristics of 4 CHAPTER 1. INTRODUCTION power series in Z[[x]] may be more apparent. In this paper, all integral domains arising as quotient rings of Z[[x]] are classified. First a brief introduction to p-adic numbers will be given, followed by some properties of power series. In particular, the irreducibility of ele- ments in Z[[x]] and the prime ideals in Z[[x]] will be discussed. Using this information, as well as the p-adic Weierstrass Preparation Theorem, the fol- lowing theorem will be proved. Main Theorem. The integral domains arising as a quotient rings of Z[[x]] are, up to isomorphism, precisely the following: Z[[x]],(Z/pZ)[[x]], Z, Z/pZ, Zp [®] where p is prime and ® is any element of Q with ® 1. Moreover, Z [®] p j jp Ç p is a local ring with maximal ideal (p,®) and is a DVR if and only if (p,®) is principal. In addition, it is also shown that for any irreducible power series f Z[[x]], 2 excluding associates of p and x, the quotient ring Z[[x]]/(f ) is isomorphic to Z [®] for some prime p, where Z is the ring of p-adic integers and ® Q is p p 2 p any root of f . 5 CHAPTER 2 Background on p-adic Numbers and Power Series Rings 2.1 p-adic Numbers Most mathematicians are familiar with Q and its completion R with respect to the usual absolute value. An absolute value on a field is defined as follows. Definition 2.1.1. An absolute value on a field K is a function from K to R 0 j¢j ¸ satisfying the following properties for any x and y in K . 1. x 0 iff x 0 j j Æ Æ 2. x y x y j j Æ j jj j 3. x y x y j Å j · j j Å j j 6 2.1. P-ADIC NUMBERS Moreover, is non-archimedean if x y max( x , y ) for all x, y K.A j ¢ j j Å j · j j j j 2 normed field is a field K together with a norm on K . j ¢ j We now introduce the p-adic absolute value. Definition 2.1.2. Let p be prime. Then is the p-adic absolute value, where j jp for all a Q, we have a is 1 divided by the power of p in the prime factoriza- b 2 j b jp a tion of b . It is not difficult to show that this definition does indeed define an abso- lute value on Q, and it is non-archimedean (see [3] or [4]). Example 2.1.3. For any prime p we have the following. 1. pn 1 j jp Æ pn 2. 1 pn j pn jp Æ 3. 16 1 j 17 j2 Æ 24 4. 16 33 j 27 j3 Æ We now give the definition of a complete normed field (see[5, 1.4-3 Defi- nition]) and the completion of a normed field (see[3, pp. 49-59]). 7 CHAPTER 2. BACKGROUND ON P-ADIC NUMBERS AND POWER SERIES RINGS Definition 2.1.4. A normed field is said to be complete if every sequence that is Cauchy with respect to the norm converges in the field. The completion j ¢ j of a normed field K with respect to the norm is a complete normed field j ¢ j containing K as a dense normed subfield and is unique up to isomorphism of normed fields. We now introduce the p-adic rational numbers. Definition 2.1.5. Let p be prime. The normed field of p-adic rational num- bers, Qp , is the completion of Q with respect to the p-adic absolute value. Note that R is the completion of Q with respect to the usual absolute value. Remark 2.1.6. In contrast to R, the series 1 p p2 Å Å Å ¢¢¢ converges in Qp . Similarly, the series 1 1 1 Å p Å p2 Å ¢¢¢ converges in R but diverges in Qp . One might ask: what are all the completions of Q with respect to its abso- lute values? The next theorem answers this question. 8 2.1. P-ADIC NUMBERS Theorem 2.1.7 (Ostrowski). Any completion of Q with respect to an absolute value is isomorphic as a normed field to R or Qp for some prime p. To help visualize elements of Q , it is useful to know that every x Q has p 2 p a unique p-adic expansion by [3, Corollary 3.3.12]. For instance every x Q 2 p can be written as follows: n0 n0 1 X n x bn0 p bn0 1p Å bn p Å Æ Å Å ¢¢¢ Æ n n0 ¸ where n Z and where 0 b p 1 for all n n . Notice that in this form, 0 2 · n · ¡ ¸ 0 it is easy to factor out the largest power of p that divides the expansion and n0 compute the p-adic absolute value of x: if b 0 then x p¡ . n0 6Æ j j Æ Definition 2.1.8. Z is the set of all x Q such that x 1. p 2 p j jp · Alternatively, Zp is the set of expansions of the following form: 2 n X1 n b0 b1p b2p bn p bn p Å Å Å ¢¢¢ Å Å ¢¢¢ Æ n 0 Æ where 0 b p 1 for all n by [3, Corollary 3.3.11]. Note that Z is a subring · n · ¡ p of Qp . Now that we have Qp , we want to find its analog to C, where C is the normed field of complex numbers. This is not as easy as going from R to C, which is a finite extension.