gh-36044: `sage.rings.number_field`: Modularization fixes, doctest cosmetics, `# needs`
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- Avoiding going through symbolic square roots in `.S_unit_solver`
- `# needs sage.symbolic`, `# needs sage.libs.*`, `# needs
sage.rings.finite_rings` etc.
- Reformatting doctests to fit on the screen
- Codestyle fixes in doctests
- Block tags for `# optional - magma`
- Handle import errors in `NumberField._pushout_`
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This is
- Part of: #29705
- Cherry-picked from: #35095
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### :hourglass: Dependencies
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URL: https://github.com/sagemath/sage/pull/36044
Reported by: Matthias Köppe
Reviewer(s): David Coudert, Kwankyu Lee, Matthias Köppe
