ProblemSet10 - 3. Show that each of the following are...

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Homework 10 Due: Monday, April 19, 2010 Problems after the double bar will not be collected but it is recommended that you do them. My assumption is that you will do them. 1. Let F be a finite field. Show that F [ x ] contains irreducible polynomials of arbitrarily high degree. 2. Factoring is only valid in an integral domain because otherwise degrees don’t add. The ring Z 6 [ x ], for example, is not an integral domain. To see that factoring does not work in a predictable way, write: (a) 5 x + 1 as a product of 2 polynomials of degree 1. (b) 5 x as a product of 2 polynomials of degree 1. (c) 2 x + 2 as a product of 2 x + 2 and a polynomial of degree 1. (d) 2 x + 2 as a product of 2 x + 2 and a polynomial of degree 2.
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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 3-3 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 eld of 81 elements. Do not just give the eld, show the construction. 6. Use long division (show it!) to nd 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 eld of order p 2 . 8. Construct a eld of 8 elements. Do not just give the eld, show the construction....
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