So-called Fundamental Theorem of Algebra

In summary, Bell (1934) argues that the fundamental theorem of algebra and its classical proof using complex variables are no longer highly valued in the field of algebra. This is in contrast to a generation ago when the theorem was considered a fundamental concept in algebra. Bell suggests that the theorem is being replaced by a different approach, closer to Kronecker's ideas. This raises questions about the traditional understanding of the fundamental theorem and its significance in algebra.
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thelema418
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This came up in one of my readings:

"Neither the so-called fundamental theorem [of algebra] itself nor its classical proof by the theory of functions of a complex variable is as highly esteemed as it was a generation ago, and the theorem seems to be on its way out of algebra to make room for something closer to what Kronecker imagined" (Bell, 1934, p. 605).

My schooling taught "the" fundamental theorem of algebra. What is different about Kronecker's treatment? What is Bell disputing when he says it is "so-called"?

Bell, E. T. (1934) The place of rigor in mathematics. The American Mathematical Monthly, 41(10), 599-607.
 
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Related to So-called Fundamental Theorem of Algebra

What is the "So-called Fundamental Theorem of Algebra"?

The "So-called Fundamental Theorem of Algebra" is a theorem in mathematics that states that every non-constant polynomial equation with complex coefficients has at least one complex root. It is considered one of the most important and fundamental theorems in algebra.

Who discovered the "So-called Fundamental Theorem of Algebra"?

The theorem was first stated by French mathematician Abraham de Moivre in the early 18th century. However, it was not until 1797 when German mathematician Carl Friedrich Gauss provided the first rigorous proof of the theorem.

Why is it called the "So-called Fundamental Theorem of Algebra"?

The term "So-called" is often used to refer to this theorem because it was originally thought to be a fundamental and necessary principle in algebra. However, with the development of more advanced mathematical concepts, it is now considered more of a fundamental result rather than a theorem that is necessary for the foundations of algebra.

What are some applications of the "So-called Fundamental Theorem of Algebra"?

The theorem has numerous applications in mathematics and other fields such as physics, engineering, and economics. It is used to solve polynomial equations, analyze complex systems, and model real-world phenomena. It also serves as the basis for many other important theorems in algebra and complex analysis.

Are there any exceptions to the "So-called Fundamental Theorem of Algebra"?

Yes, there are some exceptions to the theorem. For example, polynomials with real coefficients can have complex roots, but they can also have real roots. Additionally, there are some non-algebraic functions that do not follow the theorem, such as transcendental functions like exponential and trigonometric functions. However, the theorem still holds true for all non-constant polynomial equations with complex coefficients.

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