Algebra and number theory Fundamental theorem of algebra or
Gauss plane In his doctoral thesis from 1799, Gauss proved the
fundamental theorem of algebra which states that every non-constant single-variable
polynomial with complex coefficients has at least one complex
root. Mathematicians including
Jean le Rond d'Alembert had produced false proofs before him, and Gauss's dissertation contains a critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts led to considerable clarification of the concept of complex numbers.
Disquisitiones Arithmeticae In the preface to the
Disquisitiones, Gauss dates the beginning of his work on number theory to 1795. By studying the works of previous mathematicians like
Fermat, Euler, Lagrange, and Legendre, he realized that these scholars had already found much of what he had independently discovered. The
Disquisitiones Arithmeticae, written in 1798 and published in 1801, consolidated number theory as a discipline and covered both elementary and algebraic
number theory. Therein he introduces the
triple bar symbol () for
congruence and uses it for a clean presentation of
modular arithmetic. It deals with the
unique factorization theorem and
primitive roots modulo n. In the main sections, Gauss presents the first two proofs of the law of
quadratic reciprocity and develops the theories of
binary and ternary
quadratic forms. The
Disquisitiones include the
Gauss composition law for binary quadratic forms, as well as the enumeration of the number of representations of an integer as the sum of three squares. As an almost immediate corollary of his
theorem on three squares, he proves the triangular case of the
Fermat polygonal number theorem for
n = 3. From several analytic results on
class numbers that Gauss gives without proof towards the end of the fifth section, it appears that Gauss already knew the
class number formula in 1801. In the last section, Gauss gives proof for the
constructibility of a regular
heptadecagon (17-sided polygon) with
straightedge and compass by reducing this geometrical problem to an algebraic one. This was the first progress in regular polygon construction in over 2000 years. He shows that a regular polygon is constructible if the number of its sides is either a
power of 2 or the product of a power of 2 and any number of distinct
Fermat primes. In the same section, he gives a result on the number of solutions of certain cubic polynomials with coefficients in
finite fields, which amounts to counting integral points on an
elliptic curve. An unfinished chapter, consisting of work done during 1797–1799, was found among his papers after his death.
Further investigations One of Gauss's first results was the empirically found conjecture of 1792 – the later called
prime number theorem – giving an estimation of the number of prime numbers by using the
integral logarithm. In 1816,
Olbers encouraged Gauss to compete for a prize from the French Academy for a proof for
Fermat's Last Theorem; he refused, considering the topic uninteresting. However, after his death a short undated paper was found with proofs of the theorem for the cases
n = 3 and
n = 5. The particular case of
n = 3 was proved much earlier by
Leonhard Euler, but Gauss developed a more streamlined proof which made use of
Eisenstein integers; though more general, the proof was simpler than in the real integers case. Gauss contributed to solving the
Kepler conjecture in 1831 with the proof that a
greatest packing density of spheres in the three-dimensional space is given when the centres of the spheres form a
cubic face-centred arrangement, when he reviewed a book of
Ludwig August Seeber on the theory of reduction of positive ternary quadratic forms. Having noticed some lacks in Seeber's proof, he simplified many of his arguments, proved the central conjecture, and remarked that this theorem is equivalent to the Kepler conjecture for regular arrangements. In two papers on
biquadratic residues (1828, 1832) Gauss introduced the
ring of
Gaussian integers \mathbb{Z}[i], showed that it is a
unique factorization domain, and generalized some key arithmetic concepts, such as
Fermat's little theorem and
Gauss's lemma. The main objective of introducing this ring was to formulate the law of biquadratic reciprocity In the second paper, he stated the general law of biquadratic reciprocity and proved several special cases of it. In an earlier publication from 1818 containing his fifth and sixth proofs of quadratic reciprocity, he claimed the techniques of these proofs (
Gauss sums) can be applied to prove higher reciprocity laws.
Analysis One of Gauss's first discoveries was the notion of the
arithmetic-geometric mean (AGM) of two positive real numbers. He discovered its relation to elliptic integrals in the years 1798–1799 through
Landen's transformation, and a diary entry recorded the discovery of the connection of
Gauss's constant to
lemniscatic elliptic functions, a result that Gauss stated "will surely open an entirely new field of analysis". He also made early inroads into the more formal issues of the foundations of
complex analysis, and from a letter to Bessel in 1811 it is clear that he knew the "fundamental theorem of complex analysis" –
Cauchy's integral theorem – and understood the notion of
complex residues when integrating around
poles.
Euler's pentagonal numbers theorem, together with other researches on the AGM and lemniscatic functions, led him to plenty of results on
Jacobi theta functions, His works show that he knew modular transformations of order 3, 5, 7 for elliptic functions since 1808. Several mathematical fragments in his
Nachlass indicate that he knew parts of the modern theory of
modular forms. One of Gauss's sketches of this kind was a drawing of a
tessellation of the
unit disk by "equilateral"
hyperbolic triangles with all angles equal to \pi/4. An example of Gauss's insight in analysis is the cryptic remark that the principles of circle division by compass and straightedge can also be applied to the division of the
lemniscate curve, which inspired Abel's theorem on lemniscate division. Another example is his publication "Summatio quarundam serierum singularium" (1811) on the determination of the sign of
quadratic Gauss sums, in which he solved the main problem by introducing
q-analogs of binomial coefficients and manipulating them by several original identities that seem to stem from his work on elliptic function theory; however, Gauss cast his argument in a formal way that does not reveal its origin in elliptic function theory, and only the later work of mathematicians such as
Jacobi and
Hermite has exposed the crux of his argument. In the "Disquisitiones generales circa series infinitam..." (1813), he provides the first systematic treatment of the general
hypergeometric function F(\alpha,\beta,\gamma,x), and shows that many of the functions known at the time are special cases of the hypergeometric function. This work is the first exact inquiry into
convergence of infinite series in the history of mathematics. Furthermore, it deals with infinite
continued fractions arising as ratios of hypergeometric functions, which are now called
Gauss continued fractions. In 1823, Gauss won the prize of the Danish Society with an essay on
conformal mappings, which contains several developments that pertain to the field of complex analysis. Gauss stated that angle-preserving mappings in the complex plane must be complex
analytic functions, and used the later-named
Beltrami equation to prove the existence of
isothermal coordinates on analytic surfaces. The essay concludes with examples of conformal mappings into a sphere and an
ellipsoid of revolution.
Numerical analysis Gauss often deduced theorems
inductively from numerical data he had collected empirically. As such, the use of efficient algorithms to facilitate calculations was vital to his research, and he made many contributions to
numerical analysis, such as the method of
Gaussian quadrature, published in 1816. In a private letter to
Gerling from 1823, he described a solution of a 4x4 system of linear equations with the
Gauss-Seidel method – an "indirect"
iterative method for the solution of linear systems, and recommended it over the usual method of "direct elimination" for systems of more than two equations. Gauss invented an algorithm for calculating what is now called
discrete Fourier transforms when calculating the orbits of Pallas and Juno in 1805, 160 years before
Cooley and
Tukey found their similar
Cooley–Tukey algorithm. He developed it as a
trigonometric interpolation method, but the paper
Theoria Interpolationis Methodo Nova Tractata was published only posthumously in 1876, well after
Joseph Fourier's introduction of the subject in 1807.
Geometry Differential geometry The geodetic survey of
Hanover fuelled Gauss's interest in
differential geometry and
topology, fields of mathematics dealing with
curves and
surfaces. This led him in 1828 to the publication of a work that marks the birth of modern
differential geometry of surfaces, as it departed from the traditional ways of treating surfaces as
cartesian graphs of functions of two variables, and that initiated the exploration of surfaces from the "inner" point of view of a two-dimensional being constrained to move on it. As a result, the
Theorema Egregium (
remarkable theorem), established a property of the notion of
Gaussian curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring
angles and
distances on the surface, regardless of the
embedding of the surface in three-dimensional or two-dimensional space. The Theorema Egregium leads to the abstraction of surfaces as doubly-extended
manifolds; it clarifies the distinction between the intrinsic properties of the manifold (the
metric) and its physical realization in ambient space. A consequence is the impossibility of an isometric transformation between surfaces of different Gaussian curvature. This means practically that a
sphere or an
ellipsoid cannot be transformed to a plane without distortion, which causes a fundamental problem in designing
projections for geographical maps. A portion of this essay is dedicated to a profound study of
geodesics. In particular, Gauss proves the local
Gauss–Bonnet theorem on geodesic triangles, and generalizes
Legendre's theorem on spherical triangles to geodesic triangles on arbitrary surfaces with continuous curvature; he found that the angles of a "sufficiently small" geodesic triangle deviate from that of a planar triangle of the same sides in a way that depends only on the values of the surface curvature at the vertices of the triangle, regardless of the behaviour of the surface in the triangle interior. Gauss's memoir from 1828 lacks the conception of
geodesic curvature. However, in a previously unpublished manuscript, very likely written in 1822–1825, he introduced the term "side curvature" (German: "Seitenkrümmung") and proved its
invariance under isometric transformations, a result that was later obtained by
Ferdinand Minding and published by him in 1830. This Gauss paper contains the core of his lemma on total curvature, but also its generalization, found and proved by
Pierre Ossian Bonnet in 1848 and known as the
Gauss–Bonnet theorem.
Non-Euclidean geometry (1828) During Gauss's lifetime, the
parallel postulate of
Euclidean geometry was heavily discussed. Numerous efforts were made to prove it in the frame of the Euclidean
axioms, whereas some mathematicians discussed the possibility of geometrical systems without it. Gauss thought about the basics of geometry from the 1790s on, but only realized in the 1810s that a non-Euclidean geometry without the parallel postulate could solve the problem. In a letter to
Franz Taurinus of 1824, he presented a short comprehensible outline of what he named a "
non-Euclidean geometry", but he strongly forbade Taurinus to make any use of it. Gauss is credited with having been the one to first discover and study non-Euclidean geometry, even coining the term as well. The first publications on non-Euclidean geometry in the history of mathematics were authored by
Nikolai Lobachevsky in 1829 and
Janos Bolyai in 1832. In the following years, Gauss wrote his ideas on the topic but did not publish them, thus avoiding influencing the contemporary scientific discussion. Gauss commended the ideas of Janos Bolyai in a letter to his father and university friend Farkas Bolyai claiming that these were congruent to his own thoughts of some decades. However, it is not quite clear to what extent he preceded Lobachevsky and Bolyai, as his written remarks are vague and obscure.
Sartorius first mentioned Gauss's work on non-Euclidean geometry in 1856, but only the publication of Gauss's
Nachlass in Volume VIII of the Collected Works (1900) showed Gauss's ideas on the matter, at a time when non-Euclidean geometry was still an object of some controversy.
Early topology Gauss was also an early pioneer of
topology or
Geometria Situs, as it was called in his lifetime. The first proof of the
fundamental theorem of algebra in 1799 contained an essentially topological argument; fifty years later, he further developed the topological argument in his fourth proof of this theorem. Another encounter with topological notions occurred to him in the course of his astronomical work in 1804, when he determined the limits of the region on the
celestial sphere in which comets and asteroids might appear, and which he termed "Zodiacus". He discovered that if the Earth's and comet's orbits are
linked, then by topological reasons the Zodiacus is the entire sphere. In 1848, in the context of the discovery of the asteroid
7 Iris, he published a further qualitative discussion of the Zodiacus. In Gauss's letters of 1820–1830, he thought intensively on topics with close affinity to Geometria Situs, and became gradually conscious of semantic difficulty in this field. Fragments from this period reveal that he tried to classify "tract figures", which are closed plane curves with a finite number of transverse self-intersections, that may also be planar projections of
knots. To do so he devised a symbolical scheme, the
Gauss code, that in a sense captured the characteristic features of tract figures. In a fragment from 1833, Gauss defined the
linking number of two space curves by a certain double integral, and in doing so provided for the first time an analytical formulation of a topological phenomenon. On the same note, he lamented the little progress made in Geometria Situs, and remarked that one of its central problems will be "to count the intertwinings of two closed or infinite curves". His notebooks from that period reveal that he was also thinking about other topological objects such as
braids and
tangles. he stated the
fundamental theorem of axonometry, which tells how to represent a 3D cube on a 2D plane with complete accuracy, via complex numbers. He described rotations of this sphere as the action of certain
linear fractional transformations on the extended complex plane, and gave a proof for the geometric theorem that the
altitudes of a triangle always meet in a single
orthocenter. Gauss was concerned with
John Napier's "
Pentagramma mirificum" – a certain spherical
pentagram – for several decades; he approached it from various points of view, and gradually gained a full understanding of its geometric, algebraic, and analytic aspects. In particular, in 1843 he stated and proved several theorems connecting elliptic functions, Napier spherical pentagons, and Poncelet pentagons in the plane. Furthermore, he contributed a solution to the problem of constructing the largest-area ellipse inside a given
quadrilateral, and discovered a surprising result about the computation of area of
pentagons. == Sciences ==