abstract algebra - What is a field? - Mathematics Stack Exchange
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 sense of addition, subtraction, multiplication and division. These operations are tied together using the underlying group structure and the ...
What are the differences between rings, groups, and fields?
These axioms are identical to those of a field, except that we impose fewer requirements on the ordered pair $(R\setminus\{0\},\times)$: it now only has to be an associative structure, rather than an abelian group. Every field is a ring, and the concept of a ring can be thought of as a generalisation of the concept of a field.
real analysis - the definition of field in mathematics - Mathematics ...
Aug 15, 2023 · A field is a set F along with two operations + and ×, both commutative, such that: (F, +) is an Abelian group. If 0 is the identity of (F, +), then (F ∖ {0}, ×) is an Abelian group. × distributes over +. Alternatively, there's a structure called a ring, and a field is just a ring with some extra restrictions - mostly that everything that ...
In abstract algebra, what is an intuitive explanation for a field?
In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication.
axioms - Is $\{0\}$ a field? - Mathematics Stack Exchange
Jun 24, 2013 · 21. Often, one stipulates that 1 ≠ 0 in a field. The mathematical object you have specified doesn't really behave the way a field "should". For instance, one generally thinks of 0 as a non-invertible element, and in fact this is the case for all …
abstract algebra - Mathematics Stack Exchange
May 1, 2017 · Explain why $\mathbb{Z \times Z}$ and $\mathbb{R \times R}$ is not a field and that why any external direct sum of two fields cannot be a field. I believe it has much to do with the lack of every non-zero element having an inverse, however I am having difficulty seeing it.
When $K [x]$ is a field? - Mathematics Stack Exchange
6. If K K is the zero ring, then the polynomial K[X] K [X] is also the zero ring. The zero ring is a perfectly respectable and useful ring : it is the final object in the category of rings and corresponds to the empty scheme in algebraic geometry. But it is most emphatically not a field. Of course this is a convention, but every definition is a ...
Why is it called a skew field? - Mathematics Stack Exchange
Jan 26, 2023 · The earliest review mentioning "skew field" is for the book Finite Geometrical Systems by Levi in 1942. The term "quasi-field" is no longer used for division rings, but there was a time when "skew field" and "quasi-field" were both in use. I suspect "skew" just had the same intended meaning as "quasi", namely a skew-object is something sort of ...
Division algebra = Field? - Mathematics Stack Exchange
Nov 19, 2016 · 3. A division algebra is "essentially" an algebra with a division operation. An algebra is "essentially" a vector space with a multiplication operation. Any field is an algebra over itself (i.e. take both the scalars and vectors to be elements of the field -- the same way that any field is a vector space over itself, and then notice that ...
Complete ordered field - Mathematics Stack Exchange
9. You’re right: there is a problem with the argument, because there is a Cauchy-complete non-Archimedean ordered field, and a non-Archimedean ordered field is not complete in the sense of having least upper bounds. The standard example starts with the field F of rational functions over R, with positive cone consisting of those functions f/g ...