Fundamental Theorem of Algebra - Corollaries

Corollaries

Since the fundamental theorem of algebra can be seen as the statement that the field of complex numbers is algebraically closed, it follows that any theorem concerning algebraically closed fields applies to the field of complex numbers. Here are a few more consequences of the theorem, which are either about the field of real numbers or about the relationship between the field of real numbers and the field of complex numbers:

• The field of complex numbers is the algebraic closure of the field of real numbers.
• Every polynomial in one variable x with real coefficients is the product of a constant, polynomials of the form x + a with a real, and polynomials of the form x2 + ax + b with a and b real and a2 − 4b < 0 (which is the same thing as saying that the polynomial x2 + ax + b has no real roots).
• Every rational function in one variable x, with real coefficients, can be written as the sum of a polynomial function with rational functions of the form a/(x − b)n (where n is a natural number, and a and b are real numbers), and rational functions of the form (ax + b)/(x2 + cx + d)n (where n is a natural number, and a, b, c, and d are real numbers such that c2 − 4d < 0). A corollary of this is that every rational function in one variable and real coefficients has an elementary primitive.
• Every algebraic extension of the real field is isomorphic either to the real field or to the complex field.