What is an algebra over a field?
In mathematics, an algebra over a field is a vector space equipped with a bilinear product that satisfies certain axioms. This concept is fundamental in abstract algebra and has wide-ranging applications in various branches of mathematics and its applications. To understand what an algebra over a field is, let’s break down the key components of this definition.
Firstly, an algebra over a field is a vector space. A vector space is a collection of vectors that can be added together and multiplied by scalars, which are elements from a field. A field is a set of elements that can be added, subtracted, multiplied, and divided (except for division by zero). The most common example of a field is the set of real numbers, denoted by $\mathbb{R}$.
Secondly, an algebra over a field has a bilinear product. A bilinear product is a function that takes two vectors from the algebra and returns another vector in the algebra. The bilinear property means that the product is linear in each of its arguments. In other words, if we have vectors $a, b, c \in A$ and scalars $\alpha, \beta \in F$, then the following properties hold:
1. $(\alpha a + \beta b)c = \alpha ac + \beta bc$
2. $a(\alpha b + \beta c) = \alpha ab + \beta ac$
Lastly, the bilinear product must satisfy certain axioms. These axioms ensure that the algebraic structure is consistent and well-defined. The main axioms are:
1. Associativity: For all $a, b, c \in A$, we have $(ab)c = a(bc)$.
2. Distributivity: For all $a, b, c \in A$ and $\alpha, \beta \in F$, we have $\alpha(ab) = (\alpha a)b$ and $(\alpha + \beta)a = \alpha a + \beta a$.
An algebra over a field is a powerful tool that allows us to study the interplay between linear algebra and abstract algebra. By combining the concepts of vector spaces and bilinear products, we can explore various algebraic structures and their properties. This understanding is crucial in many areas of mathematics, including representation theory, ring theory, and algebraic geometry.
