Why is it that [tex]\mathbb{C}(x)[/tex] ([tex]\mathbb{C}[/tex] adjoined with x) is not algebraically closed? Here x is an indeterminate.(adsbygoogle = window.adsbygoogle || []).push({});

My first question is what does the field extension [tex]\mathbb{C}(x)[/tex] even mean? If E is a field extension of F, andais an transcendental element of E over F, then the notation [tex]\mathbb{C}(a)[/tex] is defined to mean the field of quotients of [tex]\mathbb{C}[a][/tex] (set of polynomials with complex coefficients). Ifais algebraic then [tex]\mathbb{C}(a)[/tex] is defined to be [tex]\mathbb{C}[a][/tex] (which is the same thing as the field of quotients of [tex]\mathbb{C}[a][/tex], since in this case [tex]\mathbb{C}[a][/tex] is a field).

Now x is an indeterminate, is that the algebraic or transcendental case? i.e. does [tex]\mathbb{C}(x)[/tex] mean field of quotients of polynomials with complex coefficients, or does it mean just polynomials with complex coefficients.

So apparently [tex]\mathbb{C}(x)[/tex] is somehow not algebraically closed. So is there a complex polynomial whose root isn't a complex number? Any help is greatly appreciated.

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# Homework Help: Problem on Algebraic Closure

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