What is meant by saying as "up to ..." in mathematical contexts When I was studying about minimal polynomials and polynomials I came across the following theorem.
Theorem:
Let $p(x) \in \mathbb{F}[x]$ be a non-constant monic polynomial. Then $p(x)$ is a unique (up to
order) product of irreducible polynomials.
Can someone please help me to understand why they say as "up to order"?
In certain other places also I've come across such phrases like "up to isomorphism" etc. What is implied by saying "up to...something"?
Thanks a lot in advance.
 A: Strictly speaking,  if something is unique "up to ...", then it is not really unique, but has several concrete instances which are nevertheless "identical from a higher point of view". This means that you have an equivalence relation and all instances belong to the same equivalence class.
For example, a factorization of a polynomial $p(x)$ is a tuple $(p_1(x),\ldots,p_n(x))$ such that $p(x)  = p_1(x) \cdot \ldots \cdot p_n(x)$. Call factorizations  $(p_1(x),\ldots,p_n(x))$ and $(q_1(x),\ldots,q_m(x))$ equivalent if $n = m$ and there exists a permutation $\pi$ of $\{1,\ldots,n\}$ such that $q_i(x) = p_{\pi(i)}(x)$. That is, the only difference is the order of the two tuples. In that sense each non-constant monic polynomial has an up to order unique factorization into irreducible polynomials - any two factorizations are equivalent.
You can also say for example that there exists an up to isomorphism unique free abelian group with one generator. In that case the equivalence relation is isomorphism of groups. The standard instance is $\mathbb Z$, but there are many other instances like the group $H$ of all group homomorphisms $\mathbb Z \to \mathbb Z$.
