📂Abstract Algebra

The Fundamental Theorem of Algebra Expressed in Terms of Abstract Algebra

Definition¹

Let $E$ be an extension field of a field $F$.

1. If every polynomial function in $F [ x ]$ has a zero in $F$, then $F$ is said to be algebraically Closed.
2. $\overline{ F_{E}} : = \left\{ \alpha \in E \mid \alpha \text{ is algebraic over } F \right\}$ is called the algebraic Closure of $F$ in $E$.

Theorem

• [1]: $F$ is algebraically closed $\iff$ every $f(x) \in F [ x ]$ factors into factors of degree $1$ in $F [ x ]$.
• [2]: For an algebraically closed $F$, there does not exist an algebraic extension field $E$ satisfying $F \lneq E$.
• [3]: The set of algebraic numbers forms a field.
• [4]: $\overline{ F_{E}}$ is a subfield of $E$.
• [5]: Every field has an algebraic closure.

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• Here, a polynomial function of course means a polynomial function excluding constant functions.

Explanation

$\overline{F}$ is the set that, when extended up to $E$, covers $F$ and takes in all the elements we can obtain algebraically. If you are somewhat familiar with topology, you can think of it in the spirit of the closed set $\overline{F} = F \cup f '$ obtained by joining $F$ with its derived set $f '$.

Through these expressions and theorems, the fundamental theorem of algebra can be stated as follows.

The field of complex numbers $\mathbb{C}$ is algebraically closed.

Unpacking this statement, by theorem [1], polynomial functions with complex coefficients factor into factors of degree $1$ in $\mathbb{C} [ x ]$, so counting multiple roots, they have exactly as many zeros as the degree of the leading term. This turns out to be equivalent to the fundamental theorem of algebra as we originally knew it.

Moreover, by theorem [2], we are guaranteed that there does not exist an algebraic extension field that contains $\mathbb{C}$ as a proper subset. This means there is no reason to consider a field larger than $\mathbb{C}$, and it is fine to take it as saying that, for all practical purposes, $\mathbb{C}$ is the largest among the fields we deal with. This fact justifies simply taking the scalar field to be the field of complex numbers when studying vector spaces as dealt with in functional analysis and elsewhere.

Proof

It is proved by complex analysis.

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