Math 361, Spring 2019, Assignment 11

Read:

1. Section 30.

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

1. Vector space (over a field $F$).

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

1. Classification of simple field extensions.

Solve the following problems:

1) Consider the field $F=\mathbb{Q}(\sqrt{2})$, the smallest subfield of $\mathbb{R}$ containing $\sqrt{2}$.

(i) Describe the elements of $F$ (i.e. give a recipe for writing down elements of $F$, and give a procedure for deciding when two such elements are equal).

(ii) Show how to add two elements of $F$ (i.e. given any two elements in the form you gave in (i), exhibit their sum in the form you gave in (i)).

(iii) Show how to multiply two elements of $F$.

2) Now consider the field $\mathbb{Q}(\pi)$, the smallest subfield of $\mathbb{R}$ containing $\pi$. Repeat steps (i)-(iii) above.

3) Finally, repeat steps (i)-(iii) for $\mathbb{Q}(\pi,\sqrt{2})$, the smallest subfield of $\mathbb{R}$ containing both $\pi$ and $\sqrt{2}$. (Hint: the hardest part is to show that the polynomial $x^2-2$ has no roots in $\mathbb{Q}(\pi)$. Do this part last.)

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