F[x]/p(x) is isomorphic to F[x]/p(x)F[x]

Dependencies:

  1. Ring isomorphism
  2. Polynomials of a ring form a ring
  3. F[x]/p(x): A ring
  4. Principal ideal
  5. Quotient Ring
  6. gH = H iff g in H
  7. Degree of sum of polynomials
  8. Polynomial division theorem

Let $F$ be a field. Let $\phi: F[x]/p(x) \mapsto F[x]/p(x)F[x]$ where $\phi(f(x)) = f(x) + p(x)F[x]$. Then $\phi$ is an isomorphism.

Proof

$\phi$ is homomorphic

\begin{align} & \phi(f_1(x)) + \phi(f_2(x)) \\ &= (f_1(x) + p(x)F[x]) + (f_2(x) + p(x)F[x]) \\ &= (f_1(x) + f_2(x)) + p(x)F[x] \tag{coset addition} \\ &= \phi(f_1(x) + f_2(x)) + p(x)F[x] \end{align}

\begin{align} & \phi(f_1(x))\phi(f_2(x)) \\ &= (f_1(x) + p(x)F[x])(f_2(x) + p(x)F[x]) \\ &= f_1(x)f_2(x) + p(x)F[x] \tag{coset multiplication} \\ &= \phi(f_1(x)f_2(x)) \end{align}

$\phi$ is injective

\begin{align} & \phi(f_1(x)) = \phi(f_2(x)) \\ &\Rightarrow f_1(x) + p(x)F[x] = f_2(x) + p(x)F[x] \\ &\Rightarrow (f_1(x)-f_2(x)) + p(x)F[x] = p(x)F[x] \tag{coset subtraction} \\ &\Rightarrow f_1(x) - f_2(x) \in p(x)F[x] \\ &\Rightarrow \exists q(x) \in F[x], f_1(x) - f_2(x) = p(x)q(x) \end{align}

\begin{align} & f_1(x), f_2(x) \in F[x]/p(x) \\ &\Rightarrow (\deg(f_1) < \deg(p) \wedge \deg(f_2) < \deg(p)) \\ &\Rightarrow \deg(f_1 - f_2) \le \max(\deg(f_1), \deg(f_2)) < \deg(p) \\ &\Rightarrow \deg(pq) < \deg(p) \\ &\Rightarrow \deg(p) + \deg(q) < \deg(p) \\ &\Rightarrow \deg(q) < 0 \\ &\Rightarrow q = 0 \\ &\Rightarrow pq = f_1 - f_2 = 0 \\ &\Rightarrow f_1 = f_2 \end{align}

$\phi$ is surjective

Let $f(x) + p(x)F[x] \in F[x]/p(x)F[x]$. By polynomial division theorem, $f = pq + r$ and $\deg(r) < \deg(p) \Rightarrow r(x) \in F[x]/p(x)$.

\begin{align} & f(x) - r(x) = p(x)q(x) \in p(x)F[x] \\ &\Rightarrow (f(x) - r(x)) + p(x)F[x] = p(x)F[x] \\ &\Rightarrow f(x) + p(x)F[x] = r(x) + p(x)F[x] = \phi(r) \end{align}

Dependency for: None

Info:

Transitive dependencies:

  1. /polynomials/commutative
  2. Group
  3. Coset
  4. For subset H of a group, a(bH) = (ab)H and (aH)b = a(Hb)
  5. Ring
  6. Polynomial
  7. Degree of product of polynomials
  8. Zero divisors of a polynomial
  9. Polynomial divisibility
  10. Degree of sum of polynomials
  11. Polynomials of a ring form a ring
  12. Ring isomorphism
  13. Integral Domain
  14. 0x = 0 = x0
  15. Ideal
  16. Field
  17. A field is an integral domain
  18. Polynomial division theorem
  19. Identity of a group is unique
  20. Subgroup
  21. Normal Subgroup
  22. Product of normal cosets is well-defined
  23. Factor group
  24. Inverse of a group element is unique
  25. gH = H iff g in H
  26. Product of ideal cosets is well-defined
  27. Quotient Ring
  28. Conditions for a subset to be a subgroup
  29. Condition for a subset to be a subgroup
  30. F[x]/p(x): A ring
  31. Conditions for a subset of a ring to be a subring
  32. Principal ideal