Author(s)

Saviour Chibeti^1*, Iness Kyapwanyama², Henry M. Phiri², Jeromy Kalunga²

¹Economics Department, University of Lusaka, Lusaka, Zambia.
²Department of Mathematics, Copperbelt University, Kitwe, Zambia.

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_p 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.

Fields, Extension Fields, Algebraic and Transcendental Extension, Algebraic Closure, Algebraically Closed Field, Absolute Value, Completion, P-Adic Field and Field of Formal Laurent Series

Chibeti, S. , Kyapwanyama, I. , Phiri, H. and Kalunga, J. (2023) An Introduction to the Theory of Field Extensions. Advances in Pure Mathematics, 13, 103-132. doi: 10.4236/apm.2023.132006.