Name:
Problem Set 7
Math 416, Section 200, Spring 2014
Due: Thursday, March 27th.
Review Sections 48 and 49 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, March 27th. Remember to fully justify all your answers,
and provide complete details.
√
1. Find all conjugates in C of 2 + i over Q.
√ √ √
2. Consider E = Q( 2, 3, 5). Compute the group of automorphisms of E, that is, list all
elements of this group, and provide the multiplication table. Is this group isomorphic to a
group you already know?
3. Let E be an algebraic extension of a field F, and let σ be an automorphism of E leaving F
fixed. Let α ∈ E, and denote f (x) = irr(α, F). Show that σ induces a permutation of the set
{β ∈ E | f (β) = 0}.
4. (20 points) Prove the following statements.
a. Let E be a field, and σ an automorphism of E. Then σ carries squares of elements of E
to squares of elements of E.
b. An automorphism of R carries positive numbers to positive numbers.
c. Let σ be an automorphism of R, and let a, b ∈ R such that a < b. Then σ(a) < σ(b).
d. The only automorphism of R is the identity automorphism. Hint: Write any real number
α as a limit of an increasing sequence of rational numbers that are smaller than α, and
also as a limit of a decreasing sequence of rational numbers that are larger than α.
5. Read the proof of Theorem 49.3 and reproduce it in your own words.
√3
6. Describe all extensions of the identity map of Q to √an isomorphism mapping Q( 2) to a
subfield of Q. Hint: You can use the fact that ω = −1+i2 3 is a primitive cubic root of unity.
7. Extra Credit. Let E be an algebraic extension of a field F. Let E be an algebraic closure of
E and F be an algebraic closure of F. Show that E and F are isomorphic.
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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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