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Consider: "Let $K$ be a field. Then every polynomial $p\in K[X]$ is transcendent over $K$." My question are:

1) How can polynomial be transcendent over something ? I thought that definition applied only for elements of $K$...

2) How can I show the above ? Do I have to find a polynomial "whose coefficients are polynomials" such that $p$ is the root of this greater polynomial ?

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Surely you mean every polynomial of degree $\ge 1$. –  Robert Israel Oct 30 '12 at 21:42
    
Elements of $K$ can't possibly be transcendental over $K$ --- only an element of an extension of $K$ (such as, for example, a polynomial) can be transcendental over $K$. –  Gerry Myerson Oct 30 '12 at 21:43

2 Answers 2

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Just think about for example the fact that $\mathbb Q(\pi)$ and $\mathbb Q[X]$ are isomorphic. They're both transcendental extensions of $\mathbb Q$.

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Hint: if $p(X)$ is a polynomial of degree $d$ with coefficients in $K$, and $Q$ is a polynomial of degree $q$ with coefficients in $K$, then $Q(p(X))$ is a polynomial of degree ...

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