Name:
Problem Set 3
Math 416, Section 200, Spring 2014
Due: Tuesday, February 11th.
Review Section 31 in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Thursday, January 23. Remember to fully justify all your answers,
and provide complete details.
√ √ √
1. Compute the degree of the field extension Q( 3, 5, 45) ≥ Q, and provide a basis. Justify
your answers.
√6
√ √3
2. Show that Q( 3, 3) = Q( 3). Hint: Read Example 31.10.
√
√
√ √
3. Show that if a and b are distinct positive rational numbers, then Q( a + b) = Q( a, b).
Hint: What is √a+1 √b ?
4. Let F, E and K be fields with F ≤ E ≤ K. Show that the following are equivalent:
a. K is algebraic over F,
b. E is algebraic over F and K is algebraic over E.
Notes. You must remember to prove both directions of the equivalence, and you must not
assume that these extensions are finite.
5. Let E be an algebraically closed extension of a field F. Show that F E (the algebraic closure of
F in E) is algebraically closed.
6. Prove that the algebraic closure of Q in C is not a finite extension. (This is an example of an
algebraic extension that is not finite.) Hint: Prove by contradiction.
7. Let R be a commutative ring with unity. Use Zorn’s Lemma to prove that if N is a proper ideal
of R, there exists a maximal ideal M of R such that N ⊆ M.
8. Extra Credit. Show that a finite field cannot be algebraically closed. Ask me for a hint when
you get stuck.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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