The basic theorem of algebra means that a polynomial with complex coefficients whose degree is not less than 1 has at least one complex root in the complex field.

This theorem actually expresses the fact that algebra in complex field is complete.

Gauss contributed the first proof of the basic theorem of algebra by using the conclusion of parameter integral; It is simpler to prove with the conclusion in the theory of complex variable functions; Rudin gave a clearer proof in his famous book Principles of Mathematical Analysis, but many theorems belonging to his book were used in the book. To understand the proof of this theorem, you have to study at least 50 pages, but the whole book has only 300 pages.

I won't go into details about the specific proof. If you are really interested, you can look up the reference materials yourself.

References:

Fichkingolz Calculus Course 14.2 [5 12] Gauss Proof of Basic Algebra Theorem Higher Education Press

Walter Rudin Theorem of Principles of Mathematical Analysis 8.8 Machinery Industry Press.

Basic Theorem of Algebra 2.5.4 What is mathematics? Basic approaches to ideas and methods, 2nd edition. Oxford, England: Oxford University Press, pp. 10 1- 103, 1996.

The basic theorem of algebra. 1. 1.7 and 3. 1.4 in the complex function manual. Boston, Massachusetts: Birkh? Users, pages 7 and 32-33, 1999.