Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
for "Arithmetical Investigations") is a textbook of
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
written in Latin by
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on the field of number theory as it not only made the field truly rigorous and systematic but also paved the path for modern number theory. In this book Gauss brought together and reconciled results in number theory obtained by mathematicians such as Fermat,
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
,
Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLegendre and added many profound and original results of his own.
Scope
The ''Disquisitiones'' covers both elementary number theory and parts of the area of mathematics now called
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
. Gauss did not explicitly recognize the concept of a group, which is central to
modern algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, so he did not use this term. His own title for his subject was Higher Arithmetic. In his Preface to the ''Disquisitiones'', Gauss describes the scope of the book as follows:
Gauss also writes, "When confronting many difficult problems, derivations have been suppressed for the sake of brevity when readers refer to this work." ("Quod, in pluribus quaestionibus difficilibus, demonstrationibus syntheticis usus sum, analysinque per quam erutae sunt suppressi, imprimis brevitatis studio tribuendum est, cui quantum fieri poterat consulere oportebat")
These sections are subdivided into 366 numbered items, which state a theorem with proof or otherwise develop a remark or thought.
Sections I to III are essentially a review of previous results, including
Fermat's little theorem
Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as
: a^p \equiv a \pmod p.
For example, if = ...
,
Wilson's theorem
In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of m ...
and the existence of primitive roots. Although few of the results in these sections are original, Gauss was the first mathematician to bring this material together in a systematic way. He also realized the importance of the property of unique factorization (assured by the fundamental theorem of arithmetic, first studied by
Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
), which he restates and proves using modern tools.
From Section IV onward, much of the work is original. Section IV develops a proof of quadratic reciprocity; Section V, which takes up over half of the book, is a comprehensive analysis of binary and ternary
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
s. Section VI includes two different
primality test
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whet ...
s. Finally, Section VII is an analysis of cyclotomic polynomials, which concludes by giving the criteria that determine which regular
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
s are constructible, i.e., can be constructed with a compass and unmarked straightedge alone.
Gauss started to write an eighth section on higher-order congruences, but did not complete it, and it was published separately after his death with the title ''Disquisitiones generales de congruentiis'' (Latin: 'General Investigations on Congruences'). In it Gauss discussed congruences of arbitrary degree, attacking the problem of general congruences from a standpoint closely related to that taken later by Dedekind, Galois, and
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing l ...
. The treatise paved the way for the theory of function fields over a
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
of constants. Ideas unique to that treatise are clear recognition of the importance of the
Frobenius morphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphi ...
, and a version of Hensel's lemma.
The ''Disquisitiones'' was one of the last mathematical works written in scholarly
Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
. An English translation was not published until 1965.
Importance
Before the ''Disquisitiones'' was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways.
The logical structure of the ''Disquisitiones'' (
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
statement followed by
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a c ...
, followed by corollaries) set a standard for later texts. While recognising the primary importance of logical proof, Gauss also illustrates many theorems with numerical examples.
The ''Disquisitiones'' was the starting point for other 19th-century European mathematicians, including Ernst Kummer, Peter Gustav Lejeune Dirichlet and
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
. Many of Gauss's annotations are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; they can now be read as containing the germs of the theories of
L-function
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may giv ...
s and complex multiplication, in particular.
The ''Disquisitiones'' continued to exert influence in the 20th century. For example, in section V, article 303, Gauss summarized his calculations of class numbers of proper primitive binary quadratic forms, and conjectured that he had found all of them with class numbers 1, 2, and 3. This was later interpreted as the determination of imaginary quadratic number fields with even discriminant and class number 1, 2, and 3, and extended to the case of odd discriminant. Sometimes called the
class number problem
In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each ''n'' ≥ 1 a complete list of imaginary quadratic fields \mathbb(\sqrt) (for negative integers ''d'') having ...
, this more general question was eventually confirmed in 1986 (the specific question Gauss asked was confirmed by
Landau
Landau ( pfl, Landach), officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990 ...
in 1902 for class number one). In section VII, article 358, Gauss proved what can be interpreted as the first nontrivial case of the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...
* Carl Friedrich Gauss, tr. Arthur A. Clarke,Not to be confused with
Arthur C. Clarke
Sir Arthur Charles Clarke (16 December 191719 March 2008) was an English science-fiction writer, science writer, futurist, inventor, undersea explorer, and television series host.
He co-wrote the screenplay for the 1968 film '' 2001: A Spac ...
, the science fiction author.S.J.: ''Disquisitiones Arithmeticae'', Yale University Press, 1965,
''Disquisitiones Arithmeticae'' (original text in Latin)
*