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:
- Subextension (of a field extension F→E).
- Subextension generated by a subset.
- Finitely generated extension.
- Simple extension.
Carefully state the following theorems (you do not need to prove them):
- Classification of simple extensions.
Solve the following problems:
- Consider the field Q(√5), the subextension of Q→R generated by the real number √5. Show that Q(√5) is isomorphic to the quotient ring Q[x]/⟨x2−5⟩. Guided by this isomorphism, write down two random elements of Q(√5), then add, subtract, multiply, and divide them.
- Following the example of the previous problem, write down two random elements of the field Q(3√2), then add, subtract, and multiply them.
- (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.
- Following the example of the previous problems, build the field Q(√5,3√2) in two stages as (Q(√5))(3√2). (The hardest part is showing that irr(3√2,Q(√5))=x3−2; if you are really stuck you may take this for granted.) Write down two random elements of Q(√5,3√2), then add, subtract, and multiply them.
- (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?
- Following the previous example, try to build the field Q(4√2,√2). Warning: irr(√2,Q(4√2)) is not x2−2. There is a factorization available.
- (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,…,αi−1)). If you ever need to know how to automate this process, follow the clues here and here.