A primitive number field is a number field 
with no intermediate field strictly
between 
and 
.
This condition is different from being generated by one element, which holds for
every number field. Every extension of prime degree
is primitive because extension degrees multiply in a tower. In a primitive number
field, every nonrational element generates the field over
.
Primitive Number Field
See also
Extension Field Degree, Number Field, Primitive ElementExplore with Wolfram|Alpha
References
Khawaja, M. and Siksek, S. "Primitive Algebraic Points on Curves." Res. Number Theory 10, 57, 2024. https://doi.org/10.1007/s40993-024-00543-4.Milne, J. S. Fields and Galois Theory. Version 5.10, Sept. 2022. https://www.jmilne.org/math/CourseNotes/FT.pdf.Cite this as:
Weisstein, Eric W. "Primitive Number Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PrimitiveNumberField.html