hw9_5_310-rem

hw9_5_310-rem - Math 310 – Section 2 Fall 2009 Remarks on...

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Unformatted text preview: Math 310 – Section 2, Fall 2009 Remarks on Homework from Chapter 5 § 5.1: Problems 3, 7, 11. § 5.2: Problems 2, 6, 14. § 5.3: Problems 2, 9. 5.1.3. 2 3 = 8; [0], [1], [ x ], [ x + 1], [ x 2 ], [ x 2 + 1], [ x 2 + x ], [ x 2 + x + 1] 5.1.11 If the nonzero polynomial p ( x ) is not irreducible, then there are polyno- mials f ( x ) ,g ( x ) ∈ F [ x ] with p ( x ) = f ( x ) g ( x ) and 0 < deg f ( x ) < deg p ( x ) and < deg g ( x ) < deg p ( x ) (by Theorem 4.10). Therefore both f ( x ) and g ( x ) are not multiples of p ( x ), and f ( x ) 6≡ 0 (mod p ( x )) and g ( x ) 6≡ 0 (mod p ( x )). But f ( x ) g ( x ) = p ( x ) ≡ 0 (mod p ( x )). 5.2.2 — 5.2.6 Every element in Q [ x ] / ( x 2- 2) has the form [ a + bx ] for a,b ∈ Q . Addition is evident: [ a + bx ] + [ c + dx ] = [( a + c ) + ( b + d ) x ]. For multiplication we get: [ a + bx ] × [ c + dx ] = [ ac + bcx + adx + bdx 2 ] = [ ac + bdx 2 +(- 2 bd +2 bd )+ adx + bcx ] = [ ac + bd ( x 2- 2) + 2 bd ) + adx + bcx ] = [ ac...
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