322final99 - K ( a, b ) ∼ = K ( x, y ). (10 points) Name...

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Math 322 Final Examination Name: May 31, 1999 Signature : 9:00–12:00 (1) Show that a polynomial f ( x ) Z [ x ] is irreducible in Z [ x ] if and only if f ( x +1) is irreducible in Z [ x ]. Using this, prove that for any odd prime number p , the p th cyclotomic polynomial Φ p ( x ) is irreducible in Z [ x ]. (15 points)
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(2) Consider R and C as vector spaces over R . Find all R -vector space homomor- phisms from R into C . (10 points)
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Name : Surname : (3) Let V be a vector space over a feld K . Let W be a subgroup oF the additive group ( V, +). ±ind necessary and su²cient conditions on W in order that the Factor group V/W become a K -vector space under the multiplication by scalars α ( v + W ) := ( αv ) + W , where α K , v V . (15 points)
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(4) Let x , y be two distinct indeterminates over a feld K . Let E be an extension oF K , and let a , b E . IF a , b are transcendental over K , prove or disprove that
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Unformatted text preview: K ( a, b ) ∼ = K ( x, y ). (10 points) Name : Surname : (5) Find all prime numbers p such that there exist exactly 3 monic irreducible polynomials of degree 2 in F p [ x ]. (10 points) (6) Is f ( x ) := x 4 + 2 x 3 + 4 x 2 + 3 x + 2 ∈ F 5 [ x ] irreducible in F 5 [ x ]. (15 points) Name : Surname : (7) Describe the elements of the Felds F 7 [ √ 3] and F 7 [ √ 5]. Are these Felds isomor-phic? If so, Fnd an isomorphism from F 7 [ √ 5] onto F 7 [ √ 3]. (15 points) (8) Consider the subfelds Q ( √ 2) and Q ( √ 3) oF R . Prove that, iF { α, β } is a Q-linearly independent subset oF Q ( √ 2), then { α, β } is Q ( √ 3)-linearly indepen-dent. (10 points)...
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This note was uploaded on 11/08/2011 for the course MATH 321 taught by Professor Ergenc during the Spring '11 term at Boğaziçi University.

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322final99 - K ( a, b ) ∼ = K ( x, y ). (10 points) Name...

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