%0 Journal Article
%T An Introduction to the Theory of Field Extensions
%A Saviour Chibeti
%A Iness Kyapwanyama
%A Henry M. Phiri
%A Jeromy Kalunga
%J Advances in Pure Mathematics
%P 103-132
%@ 2160-0384
%D 2023
%I Scientific Research Publishing
%R 10.4236/apm.2023.132006
%X 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><em>p</em>&nbsp;</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, <em>F</em>, where we can find the roots of any polynomial with coefficients in <em>F</em>.
%K Fields
%K Extension Fields
%K Algebraic and Transcendental Extension
%K Algebraic Closure
%K Algebraically Closed Field
%K Absolute Value
%K Completion
%K P-Adic Field and Field of Formal Laurent Series
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=123380