Aug 15, 2023 · As an aside, all definitions are complicated! You state that a field is a set with addition and multiplication satisfying some "complicated" axioms. In response I'd like to ask you: what is the …

Jan 4, 2012 · Division in particular is what makes a field special, separating it from, say, a ring. So the short answer to your question is: a field is an algebraic structure on a set which allows us to make …

Sep 11, 2014 · The field of complex numbers is also complete, and have the extra feature that every nonconstant polynomial can be factored in factors of degree $1$. It's said to be an algebraically …

The main difference in idea, put vaguely, is that fields are made of 'numbers' and vector spaces are made of 'collections of numbers' (vectors). You can multiply any two numbers together, and you can …

Mar 12, 2021 · Your quoted math definition of a field is one from algebra, not from analysis, but even analysts sometimes use algebra and borrow its terminology. “Physical” notions of fields are actually …

Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used?

The temperature is a scalar field: for each point in the water there is a temperature, which is a scalar, which says how hot the water is at that point. A vector field means we take some space, say a plane, …

Jan 11, 2017 · A field can be thought of as two groups with extra distributivity law. A ring is more complex: with abelian group and a semigroup with extra distributivity law. Is a ring a more basic …

May 12, 2018 · The question is about the unique (up to isomorphism) field with four elements. Assuming that such a field exists, then its addition and multiplication tables are uniquely determined by the field …

Nov 1, 2013 · The terms algebra and field indeed have different meanings in different branches of mathematics. In abstract algebra, or more precisely ring theory, a field is a commutative division ring. …