(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).
(a) Define the polynomial ring S = F2[x].
(b) Find all degree 2 irreducible polynomials. How many are there? For each,
completely describe the corresponding quadratic field extensions of F2.
(c) True of false: There is a quadratic extension of F2 that is isomorphic to Z/4Z.
Explain.
(d) Find all degree 3 irreducible polynomials. How many are there? For each,
completely describe the corresponding cubic field extensions of F2.
(e) Conjecture as to the number of, and structure of, all degree n-extensions of the
F2.
(f) Food for thought: What about for other p
How would I code these and go about responding using SAGE math program?
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