I'd' say these are pretty close. A beginner should feel very proud that they know such a useful word ^_^ It means a whole lot of things:
- a field as an open area
- a field to grow crops
- a battlefield.
- "Поле" is used in physics (eg. electromagnetic field)
- ...and mathematics (sorry, I'm not sure you know enough maths for that)
- "field of view"
- soccer field
- "margins" on a page.
I am not sure "поле" is used to describe areas of knowledge or human activity. It seems that both in Russian and in Ukranian "сфера" would be the best word for that.
The metaphor of "field testing" (practical testing or studies as opposed to purely theoretical) also works to an extent: there is the adjective "польовий". The use is different, though.
I'm regularly asking all those questions, because as a native Polish speaker learning Ukrainian through English there's high risk of things getting lost in translation.
Polish doesn't use word pole to refer to a sports field, the term boisko is used instead, so I wanted to make sure if there's a similar situation in Ukrainian.
As for field in mathematics, there are two meanings of field: (1) a function whose domain is a space, or (2) an algebraic structure supporting division, and in languages influenced by the German, those two are different: German has Feld vs Körper, Polish has pole vs ciało.
This is of course yet another reason you can't have mathematical texts translated by any random translator.
From the Ukranian wikipedia: «У футбол грають м'ячем на прямокутному трав'яному полі з воротами на двох протилежних сторонах поля».
In the mathematical sense of the word, I was talking about fields that have addition, multiplication and division defined. The whole set of axioms includes about ten of them, from a+b=b+a and the existence of 0 and 1 to a(b+c) = ab + ac and the existence of such 1/a that a · 1/a = 1. A function defined in space would be pretty boring, since it is the same as the interpretation in physics—yeah, we all know that physics uses various mathematics to get the job done.