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If f(x) ~ g(x) if and only if f and g are associates, prove this is an equivalence relation

have tried to prove this both ways, struggling

Alice
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    what do you mean associates? should we assume these are elements of a polynomial ring and this means that there is a unit such that ..? – Prahlad Vaidyanathan Oct 08 '13 at 16:35
  • associates meaning: if f(x),g(x) exist in some F[x], and f(x)=ug(x). f(x) is an associate of g(x) and u^(-1)f(x)=g(x). then f(x) and g(x) are associates – Alice Oct 08 '13 at 16:41
  • @ElizaK Please [edit] your question to include this important information. We can't read your mind after all :). Thanks in advance. – Lord_Farin Oct 08 '13 at 16:54
  • So sorry, still new at this. This is all information the professor provided, what other information should i provide? – Alice Oct 08 '13 at 16:55

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Well you need to show 3 things :

  1. Reflexivity : Take $u=1$
  2. Symmetry : If $u$ works in one direction, then $u^{-1}$ works in the other.
  3. Transitivity : If $f(x) = ug(x)$ and $g(x) = vh(x)$, then $f(x) = (uv)h(x)$, and $uv$ is a unit if $u$ and $v$ are.
Prahlad Vaidyanathan
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