A Survey on Fixed Divisors
Confluentes Mathematici, Volume 11 (2019) no. 1, p. 29-52

In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of , progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of n×n matrices over (or a Dedekind domain) could lead to the generalization of fixed divisors in that setting.

Received : 2018-05-19
Revised : 2018-11-25
Accepted : 2018-12-19
Published online : 2019-08-28
DOI : https://doi.org/10.5802/cml.54
Classification:  11Sxx,  11S05,  13F20
Keywords: Fixed divisors, Generalized factorials, Generalized factorials in several variables, Common factor of indices, Factoring of prime ideals, Integer valued polynomials
@article{CML_2019__11_1_29_0,
     author = {Devendra Prasad and Krishnan Rajkumar and A. Satyanarayana  Reddy},
     title = {A Survey on Fixed Divisors},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {11},
     number = {1},
     year = {2019},
     pages = {29-52},
     doi = {10.5802/cml.54},
     language = {en},
     url = {https://cml.centre-mersenne.org/item/CML_2019__11_1_29_0}
}
Prasad, Devendra; Rajkumar, Krishnan; Reddy, A. Satyanarayana . A Survey on Fixed Divisors. Confluentes Mathematici, Volume 11 (2019) no. 1, pp. 29-52. doi : 10.5802/cml.54. https://cml.centre-mersenne.org/item/CML_2019__11_1_29_0/

[1] David Adam Simultaneous orderings in function fields, J. Number Theory, Tome 112 (2005) no. 2, pp. 287-297 | Article | MR 2141533

[2] David Adam Pólya and Newtonian function fields, Manuscripta Math., Tome 126 (2008) no. 2, pp. 231-246 | Article | MR 2403187

[3] David Adam; Paul-Jean Cahen Newton and Schinzel sequences in quadratic fields, Actes des rencontres du CIRM, Tome 2 (2010) no. 2, pp. 15-20

[4] David Adam; Paul-Jean Cahen Newtonian and Schinzel quadratic fields, J. Pure Appl. Algebra, Tome 215 (2011) no. 8, pp. 1902-1918 | Article | MR 2776432

[5] David Adam; Jean-Luc Chabert; Youssef Fares Subsets of with simultaneous orderings, Integers, Tome 10 (2010), pp. A37, 437-451 | Article | MR 2798615

[6] David F. Anderson Elasticity of factorizations in integral domains: a survey, Factorization in integral domains (Iowa City, IA, 1996), Dekker, New York (Lecture Notes in Pure and Appl. Math.) Tome 189 (1997), pp. 1-29 | MR 1460767

[7] David F. Anderson; Paul-Jean Cahen; Scott T. Chapman; William W. Smith Some factorization properties of the ring of integer-valued polynomials, Zero-dimensional commutative rings (Knoxville, TN, 1994), Dekker, New York (Lecture Notes in Pure and Appl. Math.) Tome 171 (1995), pp. 125-142 | MR 1335709

[8] Mohamed Ayad; Rachid Bouchenna; Omar Kihel Indices in a number field, J. Théor. Nombres Bordeaux, Tome 29 (2017) no. 1, pp. 201-216 http://jtnb.cedram.org/item?id=JTNB_2017__29_1_201_0 | MR 3614524

[9] Mohamed Ayad; Omar Kihel Common divisors of values of polynomials and common factors of indices in a number field, Int. J. Number Theory, Tome 7 (2011) no. 5, pp. 1173-1194 | Article | MR 2825966

[10] Andrea Bandini Functions f:/p n /p n induced by polynomials of [X], Ann. Mat. Pura Appl. (4), Tome 181 (2002) no. 1, pp. 95-104 | Article | MR 1895028

[11] Michael Bauer Über die außerwesentlichen Diskriminantenteiler einer Gattung, Math. Ann., Tome 64 (1907) no. 4, pp. 573-576 | Article | MR 1511458

[12] Manjul Bhargava P-orderings and polynomial functions on arbitrary subsets of Dedekind rings, J. Reine Angew. Math., Tome 490 (1997), pp. 101-127 | Article | MR 1468927

[13] Manjul Bhargava Generalized factorials and fixed divisors over subsets of a Dedekind domain, J. Number Theory, Tome 72 (1998) no. 1, pp. 67-75 | Article | MR 1643292

[14] Manjul Bhargava The factorial function and generalizations, Amer. Math. Monthly, Tome 107 (2000) no. 9, pp. 783-799 | Article | MR 1792411

[15] Arnab Bose Investigations on some exponential congruences, University of Lethbridge, Canada (2016) (Masters thesis)

[16] V. Bouniakowsky Nouveaux théorèmes relatifs à la distinction des nombres premiers et à la décomposition des entiers en facteurs, Mém. Acad. Sc. St-Pétersbourg (6), Sci. Math. Phys., Tome 6 (1857), pp. 305-329

[17] Jason Boynton Pullbacks of arithmetical rings, Comm. Algebra, Tome 35 (2007) no. 9, pp. 2671-2684 | Article | MR 2356440

[18] Jason G. Boynton; Sean Sather-Wagstaff Regular pullbacks, Progress in commutative algebra 2, Walter de Gruyter, Berlin (2012), pp. 145-169 | MR 2932594

[19] Jason Greene Boynton Atomicity and the fixed divisor in certain pullback constructions, Colloq. Math., Tome 129 (2012) no. 1, pp. 87-97 | Article | MR 3007668

[20] Jakub Byszewski; Mikołaj Fraczyk; Anna Szumowicz Simultaneous p-orderings and minimizing volumes in number fields, J. Number Theory, Tome 173 (2017), pp. 478-511 | Article | MR 3581929

[21] Paul-Jean Cahen Polynômes à valeurs entières, Canad. J. Math, Tome 24 (1972), pp. 747-754

[22] Paul-Jean Cahen Polynomial closure, J. Number Theory, Tome 61 (1996) no. 2, pp. 226-247 | Article | MR 1423051

[23] Paul-Jean Cahen Newtonian and Schinzel sequences in a domain, J. Pure Appl. Algebra, Tome 213 (2009) no. 11, pp. 2117-2133 | Article | MR 2533310

[24] Paul-Jean Cahen; Jean-Luc Chabert Elasticity for integral-valued polynomials, J. Pure Appl. Algebra, Tome 103 (1995) no. 3, pp. 303-311 | Article | MR 1357791

[25] Paul-Jean Cahen; Jean-Luc Chabert Integer-valued polynomials, American Mathematical Society, Providence, RI, Mathematical Surveys and Monographs, Tome 48 (1997), xx+322 pages | MR 1421321

[26] Paul-Jean Cahen; Jean-Luc Chabert What you should know about integer-valued polynomials, Amer. Math. Monthly, Tome 123 (2016) no. 4, pp. 311-337 | Article | MR 3493376

[27] Paul-Jean Cahen; Jean-Luc Chabert Test sets for polynomials: n-universal subsets and Newton sequences, J. Algebra, Tome 502 (2018), pp. 277-314 | Article | MR 3774894

[28] Leonard Carlitz On abelian fields, Trans. Amer. Math. Soc., Tome 35 (1933) no. 1, pp. 122-136 | Article | MR 1501675

[29] Leonard Carlitz A note on common index divisors, Proc. Amer. Math. Soc., Tome 3 (1952), pp. 688-692 | Article | MR 0050627

[30] Jean-Luc Chabert On the polynomial closure in a valued field, J. Number Theory, Tome 130 (2010) no. 2, pp. 458-468 | Article | MR 2564907

[31] Jean-Luc Chabert Integer-valued polynomials: looking for regular bases (a survey), Commutative algebra, Springer, New York (2014), pp. 83-111 | MR 3330213

[32] Jean-Luc Chabert; Paul-Jean Cahen Old problems and new questions around integer-valued polynomials and factorial sequences, Multiplicative ideal theory in commutative algebra, Springer, New York (2006), pp. 89-108 | Article | MR 2265803

[33] Jean-Luc Chabert; Scott T. Chapman; William W. Smith The Skolem property in rings of integer-valued polynomials, Proc. Amer. Math. Soc., Tome 126 (1998) no. 11, pp. 3151-3159 | Article | MR 1452796

[34] Jean-Luc Chabert; Sabine Evrard On the ideal generated by the values of a polynomial, Arithmetical properties of commutative rings and monoids, Chapman & Hall/CRC, Boca Raton, FL (Lect. Notes Pure Appl. Math.) Tome 241 (2005), pp. 213-225 | Article | MR 2140695

[35] Scott T. Chapman; Barbara A. McClain Irreducible polynomials and full elasticity in rings of integer-valued polynomials, J. Algebra, Tome 293 (2005) no. 2, pp. 595-610 | Article | MR 2173716

[36] Scott T. Chapman; Vadim Ponomarenko On image sets of integer-valued polynomials, J. Algebra, Tome 348 (2011), pp. 350-353 | Article | MR 2852246

[37] Scott T. Chapman; Vadim Ponomarenko; William W. Smith Robert Gilmer’s contributions to the theory of integer-valued polynomials, Multiplicative ideal theory in commutative algebra, Springer, New York (2006), pp. 109-122 | Article | MR 2265804

[38] Zhibo Chen On polynomial functions from Z n to Z m , Discrete Math., Tome 137 (1995) no. 1-3, pp. 137-145 | Article | MR 1312448

[39] Geumlan Choi; Alexandru Zaharescu A class of exponential congruences in several variables, J. Korean Math. Soc., Tome 41 (2004) no. 4, pp. 717-735 | Article | MR 2068148

[40] Andrew M. Crabbe Generalized factorial functions and binomial coefficients, Undergraduate Honors Thesis, Trinity University, USA (2001)

[41] Leonard Eugene Dickson History of the theory of numbers. Vol. I: Divisibility and primality, Chelsea Publishing Co., New York (1966), xii+486 pages | MR 0245499

[42] D. S. Dummit; H. Kisilevsky Indices in cyclic cubic fields, Number theory and algebra, Academic Press, New York (1977), pp. 29-42 | MR 0460272

[43] Howard Theodore Engstrom On the common index divisors of an algebraic field, Trans. Amer. Math. Soc., Tome 32 (1930) no. 2, pp. 223-237 | Article | MR 1501535

[44] Sabine Evrard Bhargava’s factorials in several variables, J. Algebra, Tome 372 (2012), pp. 134-148 | Article | MR 2990004

[45] Sabine Evrard; Youssef Fares; Keith Johnson Integer-valued polynomials on lower triangular integer matrices, Monatsh. Math., Tome 170 (2013) no. 2, pp. 147-160 | Article | MR 3041736

[46] Sabine Evrard; Keith Johnson The ring of integer-valued polynomials on 2×2 matrices and its integral closure, J. Algebra, Tome 441 (2015), pp. 660-677 | Article | MR 3391941

[47] Sophie Frisch Substitution and closure of sets under integer-valued polynomials, J. Number Theory, Tome 56 (1996) no. 2, pp. 396-403 | Article | MR 1373562

[48] Sophie Frisch Polynomial separation of points in algebras, Arithmetical properties of commutative rings and monoids, Chapman & Hall/CRC, Boca Raton, FL (Lect. Notes Pure Appl. Math.) Tome 241 (2005), pp. 253-259 | Article | MR 2140698

[49] Sophie Frisch Integer-valued polynomials on algebras: a survey, Actes des rencontres du CIRM, Tome 2 (2010) no. 2, pp. 27-32

[50] Sophie Frisch A construction of integer-valued polynomials with prescribed sets of lengths of factorizations, Monatsh. Math., Tome 171 (2013) no. 3-4, pp. 341-350 | Article | MR 3090795

[51] Sophie Frisch Integer-valued polynomials on algebras, J. Algebra, Tome 373 (2013), pp. 414-425 | Article | MR 2995034

[52] Sophie Frisch Corrigendum to “Integer-valued polynomials on algebras” [J. Algebra 373 (2013) 414–425], J. Algebra, Tome 412 (2014), 282 pages | Article | MR 3215960

[53] Robert Gilmer Sets that determine integer-valued polynomials, J. Number Theory, Tome 33 (1989) no. 1, pp. 95-100 | Article | MR 1014390

[54] Robert Gilmer The ideal of polynomials vanishing on a commutative ring, Proc. Amer. Math. Soc., Tome 127 (1999) no. 5, pp. 1265-1267 | Article | MR 1473669

[55] Robert Gilmer; William W. Smith On the polynomial equivalence of subsets E and f(E) of , Arch. Math. (Basel), Tome 73 (1999) no. 5, pp. 355-365 | Article | MR 1712071

[56] Hiroshi Gunji; Donald L. McQuillan On polynomials with integer coefficients, J. Number Theory, Tome 1 (1969), pp. 486-493 | Article | MR 0249403

[57] Hiroshi Gunji; Donald L. McQuillan On a class of ideals in an algebraic number field, J. Number Theory, Tome 2 (1970), pp. 207-222 | Article | MR 0257026

[58] Richard K. Guy Unsolved problems in number theory, Springer-Verlag, New York, Problem Books in Mathematics (1994), xvi+285 pages | Article | MR 1299330

[59] Marshall Hall Indices in cubic fields, Bull. Amer. Math. Soc., Tome 43 (1937) no. 2, pp. 104-108 | Article | MR 1563496

[60] Harris Hancock Foundations of the theory of algebraic numbers. Vol. II: The general theory, Dover Publications, Inc., New York (1964), xxvi+654 pages | MR 0172865

[61] Robin Hartshorne Algebraic geometry, Springer-Verlag, New York-Heidelberg (1977), xvi+496 pages | MR 0463157

[62] Bahar Heidaryan; Matteo Longo; Giulio Peruginelli Galois structure on integral-valued polynomials, J. Number Theory, Tome 171 (2017), pp. 198-212 | Article | MR 3556682

[63] Bahar Heidaryan; Ali Rajaei Biquadratic Pólya fields with only one quadratic Pólya subfield, J. Number Theory, Tome 143 (2014), pp. 279-285 | Article | MR 3227348

[64] K. Hensel Ueber den grössten gemeinsamen Theiler aller Zahlen, welche durch eine ganze Function von n Veränderlichen darstellbar sind, J. Reine Angew. Math., Tome 116 (1896), pp. 350-356 | Article | MR 1580431

[65] István Járási Remarks on P-orderings and simultaneous orderings (Preprint)

[66] J. Latham On sequences of algebraic integers, J. London Math. Soc. (2), Tome 6 (1973), pp. 555-560 | Article | MR 0314793

[67] Amandine Leriche Pólya fields and Pólya numbers, Actes des rencontres du CIRM, Tome 2 (2010) no. 2, pp. 21-26

[68] Amandine Leriche Pólya fields, Pólya groups and Pólya extensions: a question of capitulation, J. Théor. Nombres Bordeaux, Tome 23 (2011) no. 1, pp. 235-249 http://jtnb.cedram.org/item?id=JTNB_2011__23_1_235_0 | MR 2780627

[69] Amandine Leriche Cubic, quartic and sextic Pólya fields, J. Number Theory, Tome 133 (2013) no. 1, pp. 59-71 | Article | MR 2981398

[70] Amandine Leriche About the embedding of a number field in a Pólya field, J. Number Theory, Tome 145 (2014), pp. 210-229 | Article | MR 3253301

[71] K. Alan Loper; Nicholas J. Werner Generalized rings of integer-valued polynomials, J. Number Theory, Tome 132 (2012) no. 11, pp. 2481-2490 | Article | MR 2954985

[72] Charles R. MacCluer Common divisors of values of polynomials, J. Number Theory, Tome 3 (1971), p. 33-34 | Article | MR 0268162

[73] Daniel A. Marcus Number fields, Springer-Verlag, New York-Heidelberg (1977), viii+279 pages (Universitext) | MR 0457396

[74] Donald L. McQuillan On a theorem of R. Gilmer, J. Number Theory, Tome 39 (1991) no. 3, pp. 245-250 | Article | MR 1133554

[75] Shashikant B. Mulay On integer-valued polynomials, Zero-dimensional commutative rings (Knoxville, TN, 1994), Dekker, New York (Lecture Notes in Pure and Appl. Math.) Tome 171 (1995), pp. 331-345 | MR 1335725

[76] Shashikant B. Mulay Integer-valued polynomials in several variables, Comm. Algebra, Tome 27 (1999) no. 5, pp. 2409-2423 | Article | MR 1683873

[77] Shashikant B. Mulay Polynomial-mappings and M-equivalence, J. Algebra, Tome 302 (2006) no. 2, pp. 862-880 | Article | MR 2293787

[78] M. Ram Murty; V. Kumar Murty On a problem of Ruderman, Amer. Math. Monthly, Tome 118 (2011) no. 7, pp. 644-650 | Article | MR 2826456

[79] Trygve Nagell Über zahlentheoretische Polynome, Norsk. Mat. Tidsskr, Tome 1 (1919), pp. 14-23

[80] Trygve Nagell Quelques résultats sur les diviseurs fixes de l’index des nombres entiers d’un corps algébrique, Ark. Mat., Tome 6 (1966), pp. 269-289 | Article | MR 0197443

[81] Władysław Narkiewicz Polynomial mappings, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1600 (1995), viii+130 pages | Article | MR 1367962

[82] Enric Nart On the index of a number field, Trans. Amer. Math. Soc., Tome 289 (1985) no. 1, pp. 171-183 | Article | MR 779058

[83] Alexander Ostrowski Über ganzwertige Polynome in algebraischen Zahlkörpern, J. Reine Angew. Math., Tome 149 (1919), pp. 117-124 | Article | MR 1580967

[84] Giulio Peruginelli Integer-valued polynomials over matrices and divided differences, Monatsh. Math., Tome 173 (2014) no. 4, pp. 559-571 | Article | MR 3177947

[85] Giulio Peruginelli Integral-valued polynomials over sets of algebraic integers of bounded degree, J. Number Theory, Tome 137 (2014), pp. 241-255 | Article | MR 3157790

[86] Giulio Peruginelli Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power, J. Algebra, Tome 398 (2014), pp. 227-242 | Article | MR 3123760

[87] Giulio Peruginelli Factorization of integer-valued polynomials with square-free denominator, Comm. Algebra, Tome 43 (2015) no. 1, pp. 197-211 | Article | MR 3240414

[88] Giulio Peruginelli; Nicholas J. Werner Integral closure of rings of integer-valued polynomials on algebras, Commutative algebra, Springer, New York (2014), pp. 293-305 | MR 3330225

[89] Giulio Peruginelli; Nicholas J. Werner Properly integral polynomials over the ring of integer-valued polynomials on a matrix ring, J. Algebra, Tome 460 (2016), pp. 320-339 | Article | MR 3510402

[90] Georg Pólya Über ganzwertige Polynome in algebraischen Zahlkörpern, J. Reine Angew. Math., Tome 149 (1919), pp. 97-116 | Article | MR 1580966

[91] Krishnan Rajkumar; A Satyanarayana Reddy; Devendra Prasad Semwal Fixed Divisor of a Multivariate Polynomial and Generalized Factorials in Several Variables, J. Korean Math. Soc., Tome 55 (2018) no. 6, pp. 1305-1320 | Article

[92] Mark W. Rogers; Cameron Wickham Polynomials inducing the zero function on local rings, Int. Electron. J. Algebra, Tome 22 (2017), pp. 170-186 | Article | MR 3673343

[93] Harry Ruderman; David Gale; C. Roger Glassey; G. Tsintsifas; David Shelupsky; Robert Brooks Problems and Solutions: Elementary Problems: E2468-E2473, Amer. Math. Monthly, Tome 81 (1974) no. 4, p. 405-406 | Article | MR 1537370

[94] Harry Ruderman; Carl Pomerance Problems and Solutions: Solutions of Elementary Problems: E2468, Amer. Math. Monthly, Tome 84 (1977) no. 1, p. 59-60 | Article | MR 1538254

[95] Harry Ruderman; W. Y. Velez Problems and Solutions: Solutions of Elementary Problems: E2468, Amer. Math. Monthly, Tome 83 (1976) no. 4, p. 288-289 | Article | MR 1538033

[96] Robert John Rundle Generalization of Ruderman’s Problem to Imaginary Quadratic Fields, Queen’s University, Canada (2012) (Ph. D. Thesis)

[97] Andrzej Schinzel On primitive prime factors of a n -b n , Proc. Cambridge Philos. Soc., Tome 58 (1962), pp. 555-562 | MR 0143728

[98] Andrzej Schinzel Selecta. Vol. II, European Mathematical Society (EMS), Zürich, Heritage of European Mathematics (2007), p. i-x and 859–1393 | MR 2383195

[99] Andrzej Schinzel On fixed divisors of forms in many variables. II, Analytic and probabilistic methods in number theory, TEV, Vilnius (2012), pp. 207-221 | MR 3025471

[100] Andrzej Schinzel On fixed divisors of forms in many variables, I, Math. Scand., Tome 114 (2014) no. 2, pp. 161-184 | Article | MR 3206382

[101] David Singmaster A maximal generalization of Fermat’s theorem, Math. Mag., Tome 39 (1966), pp. 103-107 | Article | MR 0197386

[102] David Singmaster On polynomial functions ( mod m), J. Number Theory, Tome 6 (1974), pp. 345-352 | Article | MR 0366881

[103] Jan Śliwa On the nonessential discriminant divisor of an algebraic number field, Acta Arith., Tome 42 (1982/83) no. 1, pp. 57-72 | Article | MR 678997

[104] Blair K. Spearman; Kenneth S. Williams Cubic fields with index 2, Monatsh. Math., Tome 134 (2002) no. 4, pp. 331-336 | Article | MR 1885929

[105] Blair K. Spearman; Kenneth S. Williams The index of a cyclic quartic field, Monatsh. Math., Tome 140 (2003) no. 1, pp. 19-70 | Article | MR 2007140

[106] Qi Sun; Ming Zhi Zhang Pairs where 2 a -2 b divides n a -n b for all n, Proc. Amer. Math. Soc., Tome 93 (1985) no. 2, pp. 218-220 | Article | MR 770523

[107] Mohammed Taous On the Pólya group of some imaginary biquadratic fields, Non-associative and non-commutative algebra and operator theory, Springer, Cham (Springer Proc. Math. Stat.) Tome 160 (2016), pp. 175-182 | MR 3613827

[108] Mohammed Taous; Abdelkader Zekhnini Pólya groups of the imaginary bicyclic biquadratic number fields, J. Number Theory, Tome 177 (2017), pp. 307-327 | Article | MR 3629246

[109] Jan Turk The fixed divisor of a polynomial, Amer. Math. Monthly, Tome 93 (1986) no. 4, pp. 282-286 | Article | MR 835297

[110] Marian Vâjâitu Estimations of the ideal generated by the values of a polynomial over a Dedekind ring, University of Bucharest, Romania (1994) (Ph. D. Thesis)

[111] Marian Vâjâitu The ideal generated by the values of a polynomial over a Dedekind ring, Rev. Roumaine Math. Pures Appl., Tome 42 (1997) no. 1-2, pp. 155-161 | MR 1650032

[112] Marian Vâjâitu An inequality involving the degree of an algebraic set, Rev. Roumaine Math. Pures Appl., Tome 43 (1998) no. 3-4, pp. 451-455 | MR 1845541

[113] Marian Vâjâitu; Alexandru Zaharescu A finiteness theorem for a class of exponential congruences, Proc. Amer. Math. Soc., Tome 127 (1999) no. 8, pp. 2225-2232 | Article | MR 1486757

[114] Robert J. Valenza Elasticity of factorization in number fields, J. Number Theory, Tome 36 (1990) no. 2, pp. 212-218 | Article | MR 1072466

[115] Vladislav V. Volkov; Fedor V. Petrov On the interpolation of integer-valued polynomials, J. Number Theory, Tome 133 (2013) no. 12, pp. 4224-4232 | Article | MR 3165637

[116] E. von Żyliński Zur Theorie der außerwesentlichen Diskriminantenteiler algebraischer Körper, Math. Ann., Tome 73 (1913) no. 2, p. 273-274 | Article | MR 1511731

[117] Bolesław Wantuła Browkin’s problem for quadratic fields, Zeszyty Nauk. Politech. Ślask. Mat.-Fiz., Tome 24 (1974), pp. 173-178 | MR 0369314

[118] Rolf Wasén On sequences of algebraic integers in pure extensions of prime degree, Colloq. Math., Tome 30 (1974), pp. 89-104 | Article | MR 0349622

[119] Nicholas J. Werner Integer-valued polynomials over quaternion rings, J. Algebra, Tome 324 (2010) no. 7, pp. 1754-1769 | Article | MR 2673759

[120] Nicholas J. Werner Integer-valued polynomials over matrix rings, Comm. Algebra, Tome 40 (2012) no. 12, pp. 4717-4726 | Article | MR 2989677

[121] Nicholas J. Werner On least common multiples of polynomials in /n[x], Comm. Algebra, Tome 40 (2012) no. 6, pp. 2066-2080 | Article | MR 2945698

[122] Nicholas J. Werner Polynomials that kill each element of a finite ring, J. Algebra Appl., Tome 13 (2014) no. 3, 1350111, 12 pages | Article | MR 3125879

[123] Nicholas J. Werner Integer-valued polynomials on algebras: a survey of recent results and open questions, Rings, polynomials, and modules, Springer, Cham (2017), pp. 353-375 | MR 3751705

[124] Melanie Wood P-orderings: a metric viewpoint and the non-existence of simultaneous orderings, J. Number Theory, Tome 99 (2003) no. 1, pp. 36-56 | Article | MR 1957243

[125] Hans Zantema Integer valued polynomials over a number field, Manuscripta Math., Tome 40 (1982) no. 2-3, pp. 155-203 | Article | MR 683038

[126] Abdelkader Zekhnini Imaginary biquadratic Pólya fields of the form (d,-2), Gulf J. Math., Tome 4 (2016) no. 4, pp. 182-188 | MR 3596398