Math 4720 & 5210 Spring 2012 -- Homework Problem Set #4

Due: Thursday, April 12th
  1. Page 264 Section 7.5 #4

  2. Identify \( \mathbb{Q}[x] / (f(x)) \). Is this quotient an integral domain? a field? when...

    Example: \( f(x)=(x^2-2)(x+1) \). Then \( \mathbb{Q}[x] / (f(x)) \cong \mathbb{Q}[x]/(x^2-2) \times \mathbb{Q}[x]/(x+1) \cong \mathbb{Q}[\sqrt{2}] \times \mathbb{Q} \). Thus this is neither an integral domain or field (the direct product of non-zero rings is never a field or integral domain).

    1. \( f(x)=(x^2+5)(x^2-1) \)
    2. \( f(x)=x^2-3 \)
    3. \( f(x)=x^2-9 \)


  3. Working in the ring \(\mathbb{Q}[x]/(f(x)) \) where \( f(x)=x^4-3x^3+x^2-x-6 \). For the following cosets \( g(x)+(f(x)) \in \mathbb{Q}[x]/(f(x)) \), run the Euclidean algorithm to find polynomials \(a(x),b(x)\in \mathbb{Q}[x] \) such that \( a(x)g(x)+b(x)f(x)=d(x) \) where \( d(x) \) is a GCD of \(f(x)\) and \(g(x)\). Then determine if \(g(x)+(f(x))\) is zero, a zero divisor, a unit, or a nobody. If it's a zero divisor, prove it. If it's a unit, find it's inverse.

    1. \( g(x)=x^3+2x-3x^2-6 \)
    2. \( g(x)=x^2-3x+2 \)
    3. \( g(x)=x^5-2x^4-2x^3-7x-6 \)


  4. Page 278-279 Section 8.1 #10

  5. Page 282 Section 8.2 #1 [5210 student also turn in #8]

  6. Page 282-283 Section 8.3 #8(a)