Math 361, Spring 2018, Assignment 8

From cartan.math.umb.edu


Read:[edit]

  1. Section 29.

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

  1. Field extension.
  2. Base field.
  3. Extension field.
  4. Injection (associated with a field extension).
  5. Algebraic element (of an extension field).
  6. Transcendental element (of an extension field).
  7. Subextension.
  8. Subextension generated by a subset.
  9. Simple extension.
  10. Finitely generated extension.
  11. Minimal polynomial (of an algebraic element; a.k.a. irr(β,F)).

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

  1. Classification of simple extensions.
  2. Theorem concerning irreducibility of minimal polynomials.

Solve the following problems:[edit]

  1. Section 29, problems 1, 3, 5, 7, 9, 11, 13, and 15.
  2. Consider the field GF(4)=Z2[x]/x2+x+1, and as usual let α denote the generator x+x2+x+1. Compute irr(α,Z2). Then compute irr(α+1,Z2).
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Questions:[edit]

Solutions:[edit]