1
$\begingroup$

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?

Thanks.

  • 1
    Note that Lang's definition of content applies to $K[x],$ not only $A[x].\:$ So that's not a problem.2012-06-03

1 Answers 1

3

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.

  • 0
    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 $K$.2012-06-03