FINAL EXAM OF ALGEBRA
Date: 2000, June 19, 15:10–17:00
An answer without reasoning will not be accepted.
1. [4%–4%–4%–5%–5%–5%]
(i) Give an example of a finite group which is not solvable.
(ii) Give an example of a field which is not perfect.
(iii) Construct a field of order 9.
(iv) Explain why Z[x] is a UFD but not a PID.
(v) Is a regular 102-gon constructible? Why?
(vi) Is Q(√ 7,√
−2) a simple extension of Q? Why?
2. [12%] Let f (x) be an irreducible polynomial over Q. Suppose that the degree of f (X) is n. Let K be the splitting field of f (X) over Q. Show that
(i) n divides [K : Q];
(ii) [K : Q] divides n!.
3. [10%] Let K be the splitting field of x4+ x2− 6 over Q. Find the Galois group G(K/Q).
4. [10%] Let p be a prime integer. Let F be a finite field of order p4and E be a finite field of order p24 such that F ≤ E. Find the Galois group G(E/F ).
5. [12%] Let K be the splitting field of the 15-th cyclotomic polynomial Φ15(x) over Q. Describe the Galois group G(K/Q) by the fundamental theorem of finitely-generated abelian groups.
6. [10%] Show that C is a finite normal extension of R. Then find the Galois group G(C/R).
7. [10%] Suppose that E is a finite normal extension of Q, and E ≤ C. Suppose that E ∩ R 6= E. Show that the order of the Galois group G(E/Q) is even.
8. [12%] Let f (x) = x4+ x + 1 ∈ Z2[x], and α be a zero of f (x). Show that the minimal polynomial of α7over Z2is x4+ x3+ 1 (i.e., irr(α7, Z2) = x4+ x3+ 1).
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