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Referring to Lang's Algebra p. 181, let $A$ be a factorial ring and $K$ its field of fractions. It is clear by definition, that the content of $f(x) \in K[x]$ is an element of $A$.

In the beginning of the paragraph above Theorem 2.1 it is mentioned that if $b \in K, \, b \neq 0$, then $cont(bf)=b \cdot cont(f)$. But then $b \cdot cont(f)$ might not be inside $A$.

Is this a typo or am i missing something?


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Note that Lang's definition of content applies to $K[x],$ not only $A[x].\:$ So that's not a problem. – Bill Dubuque Jun 3 '12 at 19:24
up vote 3 down vote accepted

With Lang's definition, I don't think that the content is necessarily an element of A. In his notation, it is perfectly fine for $ord_p a_i$ to be negative.

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I have a different edition of Lang - my page number for this is 126. Content is defined for polynomials over the field of fractions $K$ (including scalars), not over the factorial ring $A$, though factorisation in $A$ is used to define it. The concept is then deployed to draw conclusions about $A[X]$. His statement $f(X)=cf_1(X)$ with $c=cont(f)$ and $f_1(X)$ having content 1 applies over $K$. – Mark Bennet Jun 3 '12 at 19:20

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