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322second99

# 322second99 - Name Surname(5 Let A B be felds and let C be...

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Math 322 Second Midterm Examination Name: May 18, 1999 Signature : 9:30–11:00 (1) Is Q ( 3 + 5 + 7) = Q ( 3 , 5 , 7)? Why or why not? (10 points)

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(2) Find | Q ( 3 + 5) : Q ( 3) | . Justify your answer. (15 points)
Name : Surname : (3) Find the number of irreducible monic polynomials of degree 36 in F 3 [ x ], and in F 5 [ x ]. (15 points)

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(4) Construct a feld in which x 2 + x + 1 F 5 [ x ] has a root. Describe its ele- ments, explain how addition and multiplication are carried out and how inverses oF nonzero elements are Found. (20 points)

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Unformatted text preview: Name : Surname : (5) Let A , B be felds and let C be a division ring such that A ⊆ B ⊆ C , so that C is a vector space over A and over B , and B is a vector space over A . Suppose that dim A B = s ∈ N and dim B C = r ∈ N . Is it true that dim A C = rs ? (20 points) (6) Let K , E be felds and let D be an integral domain such that K ⊆ D ⊆ E . IF E is algebraic over K , does it necessarily Follow that D is a feld? (20 points)...
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322second99 - Name Surname(5 Let A B be felds and let C be...

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