Algebraic Extensions
In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Sciendo
2021-04-01
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| Series: | Formalized Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.2478/forma-2021-0004 |
| Summary: | In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over F are both finite and algebraic. We also define the field of algebraic elements of E over F and show that this field is an intermediate field of E|F. |
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| ISSN: | 1898-9934 |