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My manual is asking to prove that $\mathbb{K}(x_1, ..., x_n)$ with regards to the addition and multiplication operations is a field, which is the field of quotient field of $\mathbb{K}[x_1, ..., x_n]$.

Firstly I don't understand well the different between $\mathbb{K}(x_1, ..., x_n)$ and $\mathbb{K}[x_1, ..., x_n]$. I know that $\mathbb{K}[x_1, ..., x_n]$ is the field of polynomials with n indeterminates and coefficients over $\mathbb{K}$.

By definition a field is a commutative ring with unity in which every non-zero element is a unit.

Giuliano Malatesta
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    The ring of polynomials is not a field (What would be any inverse of $x_1$ for example ?) – GreginGre Nov 12 '20 at 11:36
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    The elements of $\mathbb K(x_1,\ldots,x_n)$ are "fractions" $p/q$ where $p,q\in\mathbb K[x_1,\ldots x_n]$, $q\ne 0$ and equality of fractions is given by $p/q=r/s\Leftrightarrow ps=qr$. The map $p\mapsto p/1$ is an injective map of $\mathbb K[x_1,\ldots,x_n]$ into $\mathbb K(x_1,\ldots,x_n)$. This is the same construction as making up $\mathbb Q$ from $\mathbb Z$. –  Nov 12 '20 at 11:38
  • This is a nice conceptual way to see why $x$ is not invertible in $R[x],,$ one that is often overlooked. – Bill Dubuque Nov 12 '20 at 12:32

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