Question

1. (Sage Exploration) In class, we primarily have worked with the field Q and its finite extensions. For each p ∈ Z primes, we can also study the field Z/pZ , which I will also denote Fp, and its finite extensions. Sage understands this field as GF(p).

   1. (a) Define the polynomial ring S = F2[x].

   2. (b) Find all degree 2 irreducible polynomials. How many are there? For each,

      completely describe the corresponding quadratic field extensions of F2.

   3. (c) True of false: There is a quadratic extension of F2 that is isomorphic to Z/4Z.

      Explain.

   4. (d) Find all degree 3 irreducible polynomials. How many are there? For each,

      completely describe the corresponding cubic field extensions of F2.

   5. (e) Conjecture as to the number of, and structure of, all degree n-extensions of the

      F2.

   6. (f) Food for thought: What about for other p

How would I code these and go about responding using SAGE math program?

Answer 1