A finite extension is a concept in field theory, a branch of abstract algebra. It refers to a situation where one field is contained within another, and the larger field has a finite dimension when viewed as a vector space over the smaller field. This means that there are only a limited number of basis elements needed to express any element of the larger field in terms of the smaller one. For example, if we take the rational numbers Q and extend them to the real numbers R, this is not a finite extension. However, extending Q to the field of rational numbers adjoined with the square root of 2, denoted as Q(√2), is a finite extension because it can be described using a finite number of basis elements.