Name:
Problem Set 8
Math 416, Section 200, Spring 2014
Due: Thursday, April 3rd.
Review Sections 50 and 51 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, April 3rd. Remember to fully justify all your answers,
and provide complete details.
√4 √6
1. Find
α
such
that
Q(
2, 2) = √Q(α). Justify
your answer by showing that your α belongs to
√6
√4 √6
4
Q( 2, 2), and by expressing 2 and 2 as polynomials in α with coefficients in Q.
2. Show that if α, β ∈ F are both separable over F then α ± β, αβ and α/β (if β , 0 are all
separable over F.
3. Let E be an algebraic extension of a field F. Show that the set of all elements in E that are
separable over F is a subfield of E. This subfield is called the separable closure of F in E.
4. Let E be an algebraic extension of a perfect field F. Prove that E is perfect.
5. Read the proof of Corollary 50.6, and reproduce it in your own words.
6. Read the proof of Corollary 50.7, and reproduce it in your own words.
7. Let α be a zero in Z2 of x3 + x2 + 1 ∈ Z2 [x]. Show that x3 + x2 + 1 splits in Z2 (α). Hint: Z2 (α)
is a finite field; find all elements in this field, and check to see which ones are the zeros you
need.
8. Show that if [E : F] = 2, then E is a splitting field over F.
9. Extra Credit.
Let f (x) ∈ F[x] and let E be the splitting field of f (x) over F. Let σ be an automorphism of E
leaving F fixed.
a. Show that σ permutes the zeros of f (x) in E.
b. Show that σ is uniquely determined by the permutation of the zeros of f (x) in E given
above.
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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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