This problem investigates the ``separableness'' of Q(3,7). You can create this number field quickly with the NumberFieldTower constructor, along with the polynomials x2−3 and x2−7. Flatten the tower with the .absolute\_field() method and use the .structure() method to retrieve mappings between the tower and the flattened version. Name the tower N and use a and b as generators. Name the flattened version L with c as a generator.
Create a nontrivial (``random'') element of L using as many powers of c as possible (check the degree of L to see how many linearly independent powers there are). Request from Sage the minimum polynomial of your random element, thus ensuring the element is a root. Construct the minimum polynomial as a polynomial over N, the field tower, and find its factorization. Your factorization should have only linear factors. Each root should be an expression in a and b, so convert each root into an expression with mathematical notation involving 3 and 7. Use one of the mappings to verify that one of the roots is indeed the original random element.
Create a few more random elements, and find a factorization (in N or in L). For a field to be separable, every element of the field should be a root of some separable polynomial. The minimal polynomial is a good polynomial to test. (Why?) Based on the evidence, does it appear that Q(3,7) is a separable extension?