Name:
Problem Set 7
Math 416, Section 500, 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.
√
2 + i over Q.
q
√
√
2. Find all conjugates in C of 1 + 2 over Q( 2).
√ √ √
3. Consider E = Q( 2, 3, 5). Consider the conjugation automorphisms
√
√
√ √
√ √
ψ √2,− √2 : (Q( √3, √5))( √2) → (Q( √3, √5))(− √2)
ψ √3,− √3 : (Q( √2, √5))( √3) → (Q( √2, √5))(− √3)
ψ √5,− √5 : (Q( 2, 3))( 5) → (Q( 2, 3))(− 5)
1. Find all conjugates in C of
√
√
a. Compute ψ √3,− √3 ◦ ψ √2,− √2 ( 2 + 3 5).
b. Find the fixed field of the automorphism ψ √5,− √5 ◦ ψ √2,− √2 .
√ √ √
4. Show that the automorphism group of E = Q( 2, 3, 5) is generated by the three automorphisms from the previous exercise. Hint: Read Example 48.17.
5. Let α be algebraic of degree n over a field F. Show that there are at most n different isomorphisms of F(α) onto a subfield of F leaving F fixed.
√ √ √
√ √
6. Let√E =√ Q( 2, 3, 5). Describe all the extensions of the identity map Q( 2, 15) →
Q( 2, 15) to an isomorphism of E onto a subfield of Q.
√3
7. 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.
8. (20 points) Extra Credit. 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.
Page 1
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.
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:
Page 2