Construct fields of each of the following orders

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SUMMARY

This discussion focuses on constructing fields of specific orders: 9, 49, 8, and 81, utilizing the polynomial representation \(\frac{F[x]}{(f(x))}\). The key theorems referenced are: (1) a polynomial \(f(x)\) in \(F[x]\) generates a field if it is irreducible, and (2) for a finite field \(F\) of order \(q\), the quotient \(\frac{F[x]}{(f(x))}\) will contain \(q^n\) elements when \(f(x)\) is of degree \(n \geq 1\). These principles are essential for constructing the desired fields.

PREREQUISITES
  • Understanding of finite fields and their properties
  • Knowledge of polynomial irreducibility in \(F[x]\)
  • Familiarity with field theory concepts
  • Basic algebraic structures and operations
NEXT STEPS
  • Study the construction of finite fields of order 9 and 49 using irreducible polynomials
  • Explore the properties of the polynomial ring \(F[x]\) and its quotient structures
  • Learn about the classification of finite fields and their applications
  • Investigate the implications of theorems on field extensions and their degrees
USEFUL FOR

This discussion is beneficial for mathematicians, algebraists, and students studying abstract algebra, particularly those interested in finite fields and polynomial theory.

mathbbb2
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Construct fields of each of the following orders: [tex]\textbf{(a)}[/tex] 9 [tex]\textbf{(b)}[/tex] 49 [tex]\textbf{(c)}[/tex] 8 [tex]\textbf{(d)}[/tex] 81 (you may exhibit these as [tex]\frac{F[x]}{(f(x))}[/tex] for some [tex]F[/tex] and [tex]f[/tex]).

Relevant Theorems to use:
[tex]\textbf{(1.)}[/tex] Let [tex]f(x)[/tex] be a polynomial in [tex]F[x][/tex]. [tex]\frac{F[x]}{(f(x))}[/tex] is a field iff [tex]f(x)[/tex] is irreducible.

[tex]\textbf{(2.)}[/tex] [tex]F[/tex] is a finite field of order [tex]q[/tex] and let [tex]f(x)[/tex] be a polynomial in [tex]F[x][/tex] of degree [tex]n \geq 1[/tex]. Then [tex]\frac{F[x]}{(f(x))}[/tex] has [tex]q^n[/tex] elements
 
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I think you should use the two theorems
 

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