if $f(x)$ is in $F[x]$. $F$ is field of integer mod $p$. $p$ is prime and $f(x)$ is irreducible over $F$ of degree $n$ . prove that $F[x]/(f(x))$ is a field with $p^n$ elements.
Problem related polynomial ring over finite field of intergers
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abstract-algebra
polynomials
ring-theory
finite-fields
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0I have approached by considering F[x] as Euclidean Ring. Also I have considered the Field formed by Quotient group of – 2012-11-17
1 Answers
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By the irreducibility of $f(x)$, the quotient $F[x]/(f(x))$ is a field.
Show that distinct polynomials in $F[x]$ of degree $
This reduces the problem to counting the number of polynomials of degree $