# Field extension

In mathematics, and, particularly, in algebra, a field extension is a pair of fields E ⊆ F , {\displaystyle E\subseteq F,} such that the operations of E are those of F restricted to E. In this case, F is an extension field of E and E is a subfield of F. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers. Field extensions are fundamental in algebraic number theory, and in the study of polynomial roots through Galois theory, and are widely used in algebraic geometry.

## Words

This table shows the example usage of word lists for keywords extraction from the text above.

Word | Word Frequency | Number of Articles | Relevance |
---|---|---|---|

extension | 5 | 13831 | 0.24 |

subfield | 3 | 359 | 0.232 |

e | 6 | 66716 | 0.212 |

field | 6 | 92745 | 0.196 |

f | 5 | 44843 | 0.193 |