Math 361, Spring 2020, Assignment 12

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

1. Subextension (of a field extension $F\rightarrow E$).
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 $\mathbb{Q}(\sqrt{5})$, the subextension of $\mathbb{Q}\rightarrow\mathbb{R}$ generated by the real number $\sqrt{5}$. Show that $\mathbb{Q}(\sqrt{5})$ is isomorphic to the quotient ring $\mathbb{Q}[x]/\left\langle x^2-5\right\rangle$. Guided by this isomorphism, write down two random elements of $\mathbb{Q}(\sqrt{5})$, then add, subtract, multiply, and divide them.
2. Following the example of the previous problem, write down two random elements of the field $\mathbb{Q}(\sqrt[3]{2})$, 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 $\mathbb{Q}(\sqrt{5},\sqrt[3]{2})$ in two stages as $\left(\mathbb{Q}(\sqrt{5})\right)(\sqrt[3]{2})$. (The hardest part is showing that $\mathrm{irr}(\sqrt[3]{2}, \mathbb{Q}(\sqrt{5}))=x^3-2$; if you are really stuck you may take this for granted.) Write down two random elements of $\mathbb{Q}(\sqrt{5},\sqrt[3]{2})$, 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 $\mathbb{Q}(\sqrt[4]{2},\sqrt{2})$. Warning: $\mathrm{irr}(\sqrt{2}, \mathbb{Q}(\sqrt[4]{2}))$ is not $x^2-2$. There is a factorization available.
7. (Optional) The previous problem shows that the most difficult part of building the extension $F(\alpha_1,\dots,\alpha_k)$ is computing the minimal polynomials $\mathbb{irr}(\alpha_i, F(\alpha_1,\dots,\alpha_{i-1}))$. If you ever need to know how to automate this process, follow the clues here and here.

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