The main theorem of algebra

The main theorem of an algebra is the statement that the field of complex numbers is algebraically closed , that is, every polynomial other than a constant (in one variable) with complex coefficients has at least one root on the field of complex numbers. The statement is also true for polynomials with real coefficients, since any real number is complex with an imaginary part zero.

There is no strictly algebraic proof of the theorem - all available ones attract non-algebraic concepts, such as the completeness of the set of real numbers or the topology of the complex plane. Moreover, the theorem is not “basic” in modern algebra - it got this name at a time when the main direction of algebra was the search for solutions to algebraic equations with real and complex coefficients.

Proof

The simplest proof of this theorem is given by complex analysis methods . We use the fact ( Liouville's theorem ) that a bounded function that is analytic on the entire complex plane ( entire function ) and has no singularities at infinity is a constant. Therefore, the function $1/p$ ${\ displaystyle 1 / p}$ $1/p$ where $p$ ${\ displaystyle p}$ $p$ - a polynomial must have at least one pole on the complex plane, and, accordingly, the polynomial has at least one root.

Corollary

An immediate consequence of the theorem is that any polynomial of degree $n$ ${\ displaystyle n}$ $n$ over the field of complex numbers has exactly $n$ ${\ displaystyle n}$ $n$ roots, taking into account their multiplicity.

Evidence of the investigation

Polynomial $f(x)$ ${\ displaystyle f (x)}$ $f(x)$ there is a root $a$ ${\ displaystyle a}$ $a$ , then, by Bezout's theorem , we represent it in the form $(x-a)g(x)$ ${\ displaystyle (xa) g (x)}$ $(x-a)g(x)$ where $g(x)$ ${\ displaystyle g (x)}$ $g(x)$ Is another polynomial. We apply the theorem to $g(x)$ ${\ displaystyle g (x)}$ $g(x)$ and we will apply it in the same way until in place $g(x)$ ${\ displaystyle g (x)}$ $g(x)$ there will not be a linear factor.

History

The theorem was first encountered by the German mathematician Peter Roth ( Peter Roth or Peter Rothe ,? -1617). In his treatise Arithmetica Philosophica (1608), he suggested that the polynomial $n$ ${\ displaystyle n}$ degree can have no more $n$ ${\ displaystyle n}$ roots. A more bold formulation was given by Albert Girard in his work “ A New Discovery in Algebra ” ( 1629 ): the equation of degree $n$ ${\ displaystyle n}$ should have exactly $n$ ${\ displaystyle n}$ roots, real (including negative ) or imaginary (the latter term refers to complex roots, the benefits of which Girard specifically specified). However, Girard made a reservation: this theorem may not be true if the "equation is incomplete", that is, some coefficients are equal to zero. The views of Roth and Girard were ahead of their time and were not widely known ^[1] .

Descartes in his work " Geometry " ( 1637 ) used the following formulation: "Every equation can have as many different roots, or values of unknown magnitude, as the last has dimensions"; further he also makes a reservation: “although one can always imagine as many roots as I said, sometimes sometimes there is not a single quantity that corresponds to these imaginary roots” ^[2] .

Maclaurin and Euler refined the statement of the theorem, giving it a form equivalent to the modern one: any polynomial with real coefficients can be decomposed into a product of linear and quadratic factors with real coefficients. D'Alembert was the first in 1746 to present a proof of this theorem, which was published in 1748 ^[3] ; however, it was based on a lemma proved only in 1851, and proved using the main theorem of algebra. In 1749, Euler's proof ^[4] was presented, and in 1751, published, while he worked on this problem almost at the same time as D'Alembert ^[5] . Also in the second half of the XVIII century there is evidence of Lagrange (1772) ^[6] , Laplace (1795) ^[7] and others. All this evidence was also based on unproven assumptions - for example, Euler considered it obvious that a real polynomial of an odd degree certainly has a real root, and Laplace suggested without proof that all the roots of the polynomial are either real or complex ^[8] .

Gauss in 1799 gave his proof, but used the same assumption as Euler; he subsequently returned to this topic more than once and gave three more proofs based on various ideas, but always involving non-algebraic means ^[8] . The first complete and rigorous proof was presented by Jean Argan in 1814; in 1816, Gauss published a rigorous proof ^[9] .

Notes

↑ History of Mathematics, Volume II, 1970 , p. 23-25.
↑ History of Mathematics, Volume II, 1970 , p. 42.
↑ D'Alembert. Recherches sur le calcul intégral // Memoires de l'academie royale des sciences et des belles lettres. - Berlin, 1748. - Vol. 2. - P. 182-224.
↑ Euler. Recherches sur les racines imaginaires des equations // Memoires de l'academie royale des sciences et belles lettres. - Berlin, 1751. - Vol. 5. - P. 222-288.
↑ Bashmakova, 1957 , p. 258.
↑ Bashmakova, 1957 , p. 259.
↑ Bashmakova, 1957 , p. 263.
↑ ¹ ² Mathematics of the 19th century. Volume I: Mathematical Logic, Algebra, Number Theory, Probability Theory / Edited by A. N. Kolmogorov , A. P. Yushkevich .. - M .: Nauka , 1978. - P. 44-49.
↑ John J. O'Connor and Edmund F. Robertson . The basic algebra theorem is a biography in the MacTutor archive.

Literature

Alekseev V. B. Abel's theorem in problems and solutions // Mathematical education . - ICMMO , 2001 .-- S. 192 . Archived on October 8, 2010.
Bashmakova I.G. On the proof of the main theorem of algebra // Historical and Mathematical Research / Edited by G.F. Rybkin, A.P. Yushkevich . - M .: State Publishing House of technical and theoretical literature, 1957. - Issue. X - S. 257-304 .
Mathematics of the XVII century // History of mathematics / Edited by A.P. Yushkevich , in three volumes. - M .: Nauka, 1970 .-- T. II.
Tikhomirov V.M., Uspensky V.V. Ten proofs of the main theorem of algebra // Mathematical education . - MCCMO , 1997. - No. 1 . - S. 50–70 .

Links

JJ O'Connor, EF Robertson. The fundamental theorem of algebra (neopr.) . MacTutor History of Mathematics archive . School of Mathematics and Statistics, University of St Andrews, Scotland (May 1996).

[_7068e2ab4910da8c-1] History of Mathematics, Volume II, 1970 , p. 23-25.

[_aaec996abc5f195c-2] History of Mathematics, Volume II, 1970 , p. 42.

[3] D'Alembert. Recherches sur le calcul intégral // Memoires de l'academie royale des sciences et des belles lettres. - Berlin, 1748. - Vol. 2. - P. 182-224.

[4] Euler. Recherches sur les racines imaginaires des equations // Memoires de l'academie royale des sciences et belles lettres. - Berlin, 1751. - Vol. 5. - P. 222-288.

[_4de8285d353f36f0-5] Bashmakova, 1957 , p. 258.

[_4de8285d353f36f1-6] Bashmakova, 1957 , p. 259.

[_4de82b5d353f3bd2-7] Bashmakova, 1957 , p. 263.

[KY44-8] ¹ ² Mathematics of the 19th century. Volume I: Mathematical Logic, Algebra, Number Theory, Probability Theory / Edited by A. N. Kolmogorov , A. P. Yushkevich .. - M .: Nauka , 1978. - P. 44-49.

[9] John J. O'Connor and Edmund F. Robertson . The basic algebra theorem is a biography in the MacTutor archive.