MIDTERM 1 FOR ALGEBRA
Date: 2000, April 17, 15:10–17:00
Each of the following problems is worth 10 points.
1.
(i) Give the definition of a field.
(ii) Give an example of a unique factorization domain but not a principal ideal domain.
2.
(i) Give the definition of a vector space over a field F .
(ii) Give an example of an infinite-dimensional vector space over R.
3.
(i) Construct a field of order 5.
(ii) Construct a field of order 25.
4. Find the greatest common divisor (in Z) of 2178, 396, 792 and 726.
5.
(i) Give the definition of an algebraic closure of a field F . (ii) Explain why C is not an algebraic closure of Q.
6. Prove that if p is a prime in an integral domain D, then p is an irreducible.
7.
(i) Show that a field is a principal ideal domain.
(ii) Show that a field is a Euclidean domain.
8.
(i) What is Z[√
−5]?
(ii) Show that 7 is an irreducible in Z[√
−5].
9. Show that if K is an algebraic extension of E and E is an algebraic extension of F , then K is an algebraic extension of F .
10.
(i) Find the degree and a basis of Q(√ 2,√
6) over Q(√ 3).
(ii) Suppose that α is a transcendental number over Q. Show that 1 + α is also transcendental over Q.
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