Dr. Marques Sophie
Office 519
Algebra 2
Fall Semester 2013
marques@cims.nyu.edu
FINAL EXAM (1h50)
Maximal Score: 200 points
Show ALL steps and make sure I understand how you get the answer to
have full credit! No material allowed!
Monday december 16th
Problems 2 and 3 both have many parts. If you are stuck on one part,
do not stop! You may assume earlier parts when working on later parts.
Problem 1: (?) 20 points Let p a odd prime number and q an integer such that
Fq is an extension of Fp . Give the answer of these questions seen in class without
justification.
1. Express q in terms of p.
2. Deduce [Fq : Fp] from the expression of q in part (1).
3. Describe Fq as the roots of some polynomial.
4. What well known group is Fq× isomorphic to?
Problem 2: (?) 60 points Let ξ = e2iπ/5 and α = cos(2iπ/5).
1. Prove that ξ 4 + ξ 3 + ξ 2 + ξ + 1 = 0.
2. Deduce that 1 + 2cos(2iπ/5) + 2cos(4iπ/5) = 0 and then that 4α2 + 2α − 1 = 0
3. Deduce that cos(2π/5) is constructible (which means that the regular pentagon
is constructible.)
4. Prove that Φ4 (x) = x4 +x3 +x2 +x+1 is irreducible in Q[x]. (Hint: First compute
Φ4 (x + 1) using the binomial formula and prove it is irreducible using a criterium
of the course, then conclude about the irreducibility of Φ4 (x).)
5. Give all the roots of φ4 (x).
6. Prove that Q(ξ) is galois over Q
7. Prove that Gal(Q(ξ), Q) ' Z/4Z and give the explicit morphism.
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Problem 3: (?) 120 points
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1. Show that the field Q( 2) is isomorphic to Q[x]/(x2 − 2).
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2. Is Q( 2) a galois extension of Q? Describe the Galois group Gal(Q( 2), Q) and
give a isomorphism to a classic group.
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3. Compute [Q( 2) : Q].
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4. Give all the proper intermediate extensions between Q( 2) and Q.
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5. √
Find a monic polynomial p(x) of degree 4 (with coefficients in Q) so that p( 2 +
5) = 0.
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6. Show that 2 − 5 ∈ Q[ 2 + 5]
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7. Deduce that Q[ 2 + 5] = Q[ 2, 5]
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8. Show that 5 is not a square in Q[ 2]
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9. Deduce that the 2 + 5 has degree 4 over Q.
10. Deduce that the polynomial found in part (a) is irreducible over Q.
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11. Deduce [Q[ 2, 5] : Q].
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12. Give a ”simple” basis for Q[ 2 + 5].
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13. Is the extension Q[ 2, 5] Galois over Q ?
14. (Bonus) Describe√it Galois
group and describe all the proper intermediate exten√
sions between Q[ 2, 5] and Q.
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(?) = easy , (??)= medium, (???)= challenge
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