sorting of number field elements is broken

[was]

The design of number field elements (and other aspects of sage) assumes that cmp defines a total ordering, which of course doesn't respect the algebraic field structure. Unfortunately, the actual implementation (in number_field_element.pyx) is buggy and doesn't define a total ordering. Look at the code and you'll see. Or just look at this example:

sage: L.<b> = NumberField(x^3-10001)
sage: b+1 < L(1667)
False
sage: L(1667) < b+1
False

I think the best correct implementation of cmp should be one that is efficient and *also* agrees with the lexicographic ordering of elements based on their representation as a polynomial in the generator of the number field. I did implement this as part of the patch bomb #9541. The point of the present ticket is to "backport" something like this out of #9541, or implement a new fix from scratch. This is motivated by #9400.

[jdemeyer]

See also #6132, there is some code to compare number field elements.

[was]

I'm closing this as a dup of #6132.