Undergraduate Algebra pp 158-183 | Cite as

# Field Theory

Chapter

## Abstract

Let with coefficients in

*F*be a field. An element α in some extension of*F*is said to be**algebraic**over*F*if there exists a non-zero polynomial*f*∈*F*[*t*] such that*f*(α) = 0, i.e. if α satisfies a polynomial equation$$ {{\text{a}}_{{\text{n}}}}{\alpha ^{{\text{n}}}} + {\text{ }}...{\text{ }} + {\text{ }}{{\text{a}}_{0}} = {\text{ }}0 $$

*F*, not all 0. If*F*is a subfield of*E*, and every element of*E*is algebraic over*F*, we say that*E*is**algebraic**over*F*.## Keywords

Rational Number Symmetric Group Galois Group Galois Extension Irreducible Polynomial
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag New York Inc. 1987