Field and Algebra What is the difference between "algebra" and "field"? In term of definition in Abstract algebra.
(In probability theory, sigma-algebra is a synonym of sigma-field, does this imply 
algebra is the same as field?)
 A: An algebra over a field is like a vector space with some sort of multiplication between vectors, like 3-dimensional real space with the cross product.
A field is like a set with some notion of addition, subtraction, multiplication and division, like the field of real numbers.
Every field is an algebra because every field is a (one dimensional) vector space, but not every algebra is a field. The previous example of real 3-dimensional space with the cross product is such an algebra.
A: An algebra is a ring that has the added structure of a field of scalars and a coherent (see below) multiplication. Some examples of algebras:


*

*M_n(F), where $F$ is any field.

*$C(T)$, continuous real (or complex)-valued functions on a topological space $T$ (here the scalars could be either the real or the complex numbers).

*$B(X)$, bounded operators over a Banach space $X$, with complex (or real) scalars. 

*$F[x]$, polynomials over a field $F$. 


A field, on the other hand, is a commutative ring where every nonzero elements is invertible (i.e. a commutative division ring). 
Note: "coherent multiplication" means that given $x,y$ in the algebra and $\alpha,\beta$ in the field, 
$$
\alpha(x+y)=\alpha x+\alpha y,\ \ (\alpha+\beta)x=\alpha x+\beta x, \ \ (\alpha\beta)x=\alpha(\beta x),\ \ (\alpha x)y=x(\alpha y). 
$$
