Unformatted text preview: 3. Show that each of the following are irreducible over Q . (a) x 5 + 9 x 4 + 12 x 2 + 6 (b) x 33 x + 3 (c) x 3 + 2 x 2 + 4 x + 1 (d) x 3 + 3 x 2 + 2 (e) x 5 + 5 x 2 + 1 4. Let p be a prime. Determine the number of polynomials of the form x 2 + ax + b which are irreducible over Z p . 5. Construct a ﬁeld of 81 elements. Do not just give the ﬁeld, show the construction. 6. Use long division (show it!) to ﬁnd the quotient and remainder when 3 x 3 + x 2 + 2 is divided by 2 x + 3 in Z 5 [ x ]. 7. Show that for every prime p there exists a ﬁeld of order p 2 . 8. Construct a ﬁeld of 8 elements. Do not just give the ﬁeld, show the construction....
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 Spring '11
 Johnson
 Algebra, Polynomials, Division, Integral domain, arbitrarily high degree

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