hi there i was looking through my lecture notes and i'm struggling to understand a particular piece of notation the vertical line | and i was wondering if you could explain its meaning

$$f \sim g \iff f - g \text{ is an element of } (x^2) \iff x^2|f-g$$

where $f$ and $g$ are elements of polynomial ring $R[x]$ and $(x^2)$ is an ideal s.t. $\{f\cdot x^2 \text{ is an element of } R[x] \mid f \text{ is an element of } R[x]\}$

see i understand the second use of | but not the first. could anyone explain the meaning to me please

  • 1
    $\begingroup$ What is the first use of $|$? That $x^2\mid f(x)-g(x)$? This means that $x^2$ divides $f(x)-g(x)$. $\endgroup$ – Dietrich Burde Mar 2 '18 at 19:15
  • $\begingroup$ The first means "divides evenly". $\endgroup$ – lulu Mar 2 '18 at 19:15
  • $\begingroup$ @DietrichBurde Oh, I just meant "divides". In English we often say things like $3$ divides $9$ evenly, to distinguish it from situations like $\frac 83\in \mathbb Q$. $\endgroup$ – lulu Mar 2 '18 at 19:18
  • $\begingroup$ I see, "a divides b evenly", e.g., here. I was confused by the word "even" (to divide oddly) $\endgroup$ – Dietrich Burde Mar 2 '18 at 19:18
  • 1
    $\begingroup$ @DietrichBurde exactly. Here "evenly" has nothing to do with parity...it's "even" in the sense of uniform or fair. As in, "you have a bunch of toys and you wish to distribute them to the kids evenly." $\endgroup$ – lulu Mar 2 '18 at 19:19

The first use means 'divides' — thus, there is a polynomial $h(x)$ such that $f(x)-g(x) = x^2h(x)$.

Also, to ensure proper spacing use the \mid command; e.g., $x^2 \mid f(x) - g(x)$.

Finally, since $x$ is being used as a variable, it's a good idea to write $f(x)$ instead of $f$ in this context since the polynomial $x^2$ doesn't have a name.

  • $\begingroup$ I prefer to use the colon : instead of the vertical line in set-notation, especially if I've already used it for "divides". $\endgroup$ – DanielWainfleet Mar 3 '18 at 2:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.