**Michele Cipolla**'s parents were Luigi Cipolla and Rosalia Moncada. He attended secondary school in Palermo, following which he entered the Scuola Normale in Pisa. In Pisa he was taught by several outstanding mathematicians but was particularly attracted to the courses given by Luigi Bianchi and Ulisse Dini. He attended Bianchi's lectures on number theory, and his lectures on group theory and Galois theory, which strongly influenced him to undertake research in algebra. After two years in Pisa, Cipolla returned to Palermo where he studied for his doctorate at the university under Gabriele Torelli's supervision. Torelli (1849-1931) had been appointed as Professor of Algebra at Palermo in 1891 and played a major role in applying the theory of functions to the study of prime numbers. He had received the mathematical prize of the Naples Academy in 1899 for his work on the totality of primes. Cipolla was awarded his doctorate from Palermo in 1902 for his thesis on asymptotic determinations of primes. He published

*La determinazione assintotica dell'n imo numero primo*in 1902,

*Un metodo per la risolutione della congruenza di secondo grado*in 1903, and

*Sui numeri composti P, che verificanola congruenza di Fermat*

*a*

^{P-1}= 1 (mod

*P*) in

*Annali di Matematica*in 1904. This contains his famous results on infinitely many pseudoprimes to a given base

*a*.

When he had been in Pisa, Cipolla had been introduced to group theory in Bianchi's lectures. While at Palermo, although doing research for his doctorate in number theory, he attended the seminars led by Francesco Gerbaldi in which he introduced students to the latest results in mathematics including group theory. Although an extremely productive mathematician, in 1904 Cipolla became a secondary school teacher in Corleone, a small town about 35 km south of Palermo. There he lived in the house of a Mr Lisi whose two sisters also lived there [9]:-

Except for walking with Lisi and another tenant who lived in the house, and occasionally being invited for coffee in the afternoon with Mr Lisi's sisters, he stayed in his room working on mathematical research.The house was comfortable, secluded, and there Cipolla found the atmosphere and the serenity necessary to carry out some of his research.

He taught at the Regio Ginnasio "G Baccelli" in Corleone until 1911 but, despite devoting much time to his teaching duties, he was very active in mathematical research, doing some of his most creative work during these years. He published papers such as *Estensione delle formole di Meissel-Rogel e di Torelli sulla totalità dei numeri primi che non superano un numero assegnato* (1905), *Sulla risoluzione apiristica delle congruenze binomie secondo un modulo primo* (1906), *Sulla teoria dei gruppi abeliani* (1908), and *Sulla struttura dei gruppi d'ordine finito* (1909). Laura Martini writes [5]:-

In 1911 Cipolla was appointed as Professor of Algebraic Analysis at the University of Catania, in the town of Catania on the east coast of Sicily. One of his publications during this period wasCipolla was one of the most remarkable Italian algebraists in his days. His scientific interests covered several fields of study, among them, number theory, the theory of finite groups, analysis, foundations, and the history and pedagogy of mathematics. He published some one hundred works in addition to a series of texts written for secondary school and a number of remarkable university-level treatises. Between1909and1912, in particular, he defined the notion of fundamental subgroups of a group and published his results in four memoirs in the 'Rendiconti della Accademia delle Scienze di Napoli'.

*Sulle equazioni algebriche le cui radici sono tutte radici dell'unità*(1914) and another was the innovative monograph

*Analisi algebraica ed introduzione al calcolo infinitesimale*. Ernst Jacobsthal writes in a review:-

From 1916, Cipolla was joined in Catania by Gaetano Scorza who also worked on group theory and had generalised some of Cipolla's work on finite groups to infinite groups. Cipolla's most famous publication on group theory is his three-volume treatiseA book that contains much more than its title promises.

*Teoria dei gruppi di ordine finito e sue applicazioni*which appeared between 1919 and 1923 [5]:-

In 1921 a new periodical was started by the Circolo Matematico di Catania calledThe work includes the theory of abstract groups, the theory of groups of substitutions, and Galois's theory of algebraic equations. In it, Cipolla gathered the lectures on group theory and Galois theory he had given at the University of Catania in the academic years1920-1921and1921-1922, and it can be considered the most complete treatise written in the Italian language on classical Galois theory.

*Esercitazioni Matematiche*. It was intended to be read by university students and was edited by Cipolla. He published a number of articles in this periodical including

*Nulla e zero*in 1937. Albert A Bennett writes in a review:-

In 1923, the year in which the third volume of his group theory treatise was published, Cipolla returned to Palermo where he was appointed as professor. He remained in this post for the rest of his life. He continued to publish work on group theory with papers such asThe author describes the notion off the null class and discusses its introduction and the symbols used to represent it, in the history of symbolic logic. The number zero "late child of nothing" is treated historically in connection with algebra and the digit zero is considered in connection with place notation among the classical writers. The important philosophical value of the number zero for mathematics and physical sciences is stressed.

*I sottogruppi fondamentali di un gruppo di Hölder*(1926), and on finite fields with papers such as

*Formule di risoluzione apiristica delle equazioni di grado qualunque in un corpo finito*(1930). His textbook

*La matematica elementare nei suoi fondamenti, nei riguardi didattici e negli sviluppi superiori*proved very popular. It was first published in 1927 with a second edition in 1929, and finally a third edition appeared in 1949 after Cipolla's death. With his student Gaspare Mignosi (1875-1951), who worked with him in Catania and determined the fundamental subgroups of the linear projective group of dimension two over a finite field, Cipolla wrote secondary school level textbooks such as

*Analisi matematiche elementare*(1924),

*Lezioni di Algebra per il liceo classico, per il biennio del liceo scientifico e dell'istituto tecnico nautico*and

*Geometria elementare per Ginnasi Superiori e i Licei : vol. 1 (per il Ginnasio superiore) : vol. 2 - Pei Licei : preparazione alla prova scritta di Matematica negli Esami di Maturità Scientifica*. Also with Vincenzo Amato (1881-1963), another of his students in Catania who studied the properties of those algebraic equations whose Galois group was the fundamental subgroup of the whole group, he wrote secondary school texts such as

*Algebra elementare : per il ginnasio superiore e per le classi 3 e 4 dell'istituto magistrale inferiore*(1926), and

*Aritmetica prattica per le scuole industriale. Vol. I*(1927). Among his other books we mention

*Lezioni di calcolo infinitesimale*(1925), and we mention that he also wrote historical articles such as

*Evaristo Galois nel primo centenario della sua morte*(1934), and

*Il contributo italiano alla renascita della matematica nel Duecento*(1935).

Cipolla received many honours for his outstanding contributions to mathematics and mathematical education, such as being elected Vice President of the Circolo Matematico di Palermo. He was also elected a member of the Accademia Gioenia di Catania, the Pontaniana di Napoli and the Accademia dei Lincei. He has also been honoured with the 'Liceo Scientifico Statale Michele Cipolla' in Castelvetrano, Sicily being dedicated to him. Also in Castelvetrano the 'Via Michele Cipolla' is named for him, while in Palermo there is a 'Via Michele Cipolla' on which the Universita' Degli Studi Di Palermo is situated.

Finally let us quote from Guido Zappa and Giovanni Zacher [2]:-

Nearly half a century after his death, Michele Cipolla seems like a very significant personality in Italian mathematics in the first half of this century. At a time when, in our country, Algebra and Number Theory were little studied, he managed to create a school which, although geographically limited, helped to keep alive interest in these disciplines.

**Article by:** *J J O'Connor* and *E F Robertson*