[Basic Theorem of Algebra] means that for a complex field, the complex coefficient of at least one polynomial is not less than 1. It is concluded that a polynomial with complex coefficients of degree n has only n roots in the complex number field, and the multiple roots are calculated according to the multiplicity.

The original idea of this theorem is that Indian mathematician Bashgaro [114-1185? ] was put forward in 1 150. He put forward a formula for finding the roots of a quadratic equation with one variable, and found the possibility of negative numbers as the roots of the equation. He began to contact the number of roots of the equation, that is, the quadratic equation with one variable has two roots. Pu Skaro called this idea [Lilavati], which originally meant "beauty" and was also his daughter's name.

1629, Girard, a Dutch mathematician, put forward his conjecture in "New Discovery of Algebra" and asserted that the polynomial equation of degree n has n roots, but he did not give proof.

1637, Descartes [1596- 1650] proposed in the third volume of Geometry that an equation with many degrees has many roots, including imaginary roots and negative roots that he did not admit.

Euler clearly pointed out in his letter to a friend in February 1742 15 that polynomials with any real coefficients can be decomposed into products of first and second factors. D'Alembert, Lagrange and Euler all tried to prove this theorem, but unfortunately the proof was incomplete. Gauss gave the first substantial proof in 1799, but it is still not strict. Later, he gave three other proofs [18 14- 15, 18 16, 1848- 1850], Basic Theorem of Algebra.

Gauss's method of studying the basic theorem of algebra opens up a new way for exploring the existence problem in mathematics. Before the 20th century, the research objects of algebra were all based on real number fields or complex number fields, so the basic theorems of algebra at that time played a core role.