This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to al...This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field F<sub>p </sub>using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.展开更多
We define a metric that makes the algebraic closure of a finite field F_(p) into a UDBG(uniformly discrete with bounded geometry)metric space.This metric stems from algebraic properties of F_(p).From this perspective,...We define a metric that makes the algebraic closure of a finite field F_(p) into a UDBG(uniformly discrete with bounded geometry)metric space.This metric stems from algebraic properties of F_(p).From this perspective,for F_(p)we explore common research themes in metric spaces,reveal how peculiar properties naturally arise,and present it as a new type of example for certain well-studied questions.展开更多
文摘This paper unfolds and reviews the theory of abstract algebra, field extensions and discusses various kinds of field extensions. Field extensions are said to be algebraic or transcendental. We pay much attention to algebraic extensions. Finally, we construct finite extensions of Q and finite extensions of the function field over finite field F<sub>p </sub>using the notion of field completion, analogous to field extensions. With the study of field extensions, considering any polynomial with coefficients in the field, we can find the roots of the polynomial, and with the notion of algebraically closed fields, we have one field, F, where we can find the roots of any polynomial with coefficients in F.
文摘We define a metric that makes the algebraic closure of a finite field F_(p) into a UDBG(uniformly discrete with bounded geometry)metric space.This metric stems from algebraic properties of F_(p).From this perspective,for F_(p)we explore common research themes in metric spaces,reveal how peculiar properties naturally arise,and present it as a new type of example for certain well-studied questions.
