This theorem was first given by Girard, a Dutch mathematician, in his book New Discovery of Algebra (1629). He speculated and asserted that a polynomial equation of degree n has n roots, but he did not give proof. Descartes also put forward this theorem in 1637, but its expression is different from modern times. Kraulin and Euler made the expression of this theorem more accurate. A form equivalent to modern expression is given: any polynomial with real coefficients can be decomposed into the product of the first and second factors of real coefficients. D'Alembert gave the first proof of the basic theorem of algebra in 1746. By the second half of the18th century, Euler, La Silas, Lagrange and others had given some proofs one after another. All these proofs presuppose that some "ideal" roots of polynomials do exist. Then prove that at least one of these roots is a complex number. In 1799, Gauss gave the first substantial proof that the roots of polynomials do not exist, but it is still not strict enough. Later, he gave three other proofs (1814-1815,65438). 1848- 1850). 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.