Math 361, Spring 2020, Assignment 12

From cartan.math.umb.edu


Carefully define the following terms, then give one example and one non-example of each:

  1. Subextension (of a field extension FE).
  2. Subextension generated by a subset.
  3. Finitely generated extension.
  4. Simple extension.

Carefully state the following theorems (you do not need to prove them):

  1. Classification of simple extensions.

Solve the following problems:

  1. Consider the field Q(5), the subextension of QR generated by the real number 5. Show that Q(5) is isomorphic to the quotient ring Q[x]/x25. Guided by this isomorphism, write down two random elements of Q(5), then add, subtract, multiply, and divide them.
  2. Following the example of the previous problem, write down two random elements of the field Q(32), then add, subtract, and multiply them.
  3. (Optional) Try to divide the elements you wrote down in the previous problem. If you are stuck, see https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Simple_algebraic_field_extensions.
  4. Following the example of the previous problems, build the field Q(5,32) in two stages as (Q(5))(32). (The hardest part is showing that irr(32,Q(5))=x32; if you are really stuck you may take this for granted.) Write down two random elements of Q(5,32), then add, subtract, and multiply them.
  5. (Optional) Try to divide the elements you wrote down above. In fact this is extremely tedious; can you imagine how you might get a computer to do this for you?
  6. Following the previous example, try to build the field Q(42,2). Warning: irr(2,Q(42)) is not x22. There is a factorization available.
  7. (Optional) The previous problem shows that the most difficult part of building the extension F(α1,,αk) is computing the minimal polynomials irr(αi,F(α1,,αi1)). If you ever need to know how to automate this process, follow the clues here and here.
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