Definition:Generator of Field
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Definition
Let $F$ be a field.
Let $S \subseteq F$ be a subset and $K \le F$ a subfield.
The field generated by $S$ is the smallest subfield of $F$ containing $S$.
The subring of $F$ generated by $K \cup S$, written $K \sqbrk S$, is the smallest subring of $F$ containing $K \cup S$.
The subfield of $F$ generated by $K \cup S$, written $\map K S$, is the smallest subfield of $F$ containing $K \cup S$.
Also see
- Definition:Generator of Field Extension
- Definition:Generated Field Extension
- Subfield Generated by Subfield and Set equals Generated Field Extension
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $23$. The Field of Rational Numbers