What is meant by saying as "up to ..." in mathematical contexts

Question

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.

It means that if $p(x)$ has $2$ representations as a product of irreducible polynomials then their irreducible factors are the same, the only difference might be their order. Another example: "there is one group of order $11$ up to isomorphism" means that any two groups of order $11$ are isomorphic. And so on... — Mark, Oct 15 '20 at 16:44
With fixed irreducible polynomials $f(x), g(x)$, $~p(x) = f(x) \times g(x)$ is not to be considered a distinct representation from $p(x) = g(x) \times f(x).$ — user2661923, Oct 15 '20 at 16:50

score 3 · Accepted Answer · edited Oct 15 '20 at 17:38

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 $\mathbb Z \to \mathbb Z$.

Thank you very much @PaulFrost it is very useful :) – Buddhini Angelika Oct 16 '20 at 06:14 — Buddhini Angelika, Oct 16 '20 at 06:14

What is meant by saying as "up to ..." in mathematical contexts

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