Z[x]/(1-x)/(x^{p-1}+ +x+1) = Z/pZ ?

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SUMMARY

The discussion centers on the equivalence of the ring quotient

Z[x]/(1-x)/(x^{p-1}+\cdots+x+1)

and the finite field

Z/pZ

, where

p

is a prime and

x=\zeta_p

represents a primitive pth root of unity. The participants clarify that setting

1-x=0

leads to

Z[x]/(1-x)=Z

, which simplifies the expression to

Z/(p)

. The confusion arises from the interpretation of the polynomial degree in the denominator, specifically whether it is

x^{p-1}+\cdots+x+1

or

x^{p-2}+\cdots+x+1

.

PREREQUISITES

  • Understanding of ring theory and polynomial rings
  • Knowledge of finite fields, specifically

    Z/pZ

  • Familiarity with roots of unity and their properties
  • Basic algebraic manipulation of polynomial expressions

NEXT STEPS

  • Study the properties of

    primitive roots of unity

    in algebraic structures
  • Learn about

    ring quotients

    and their applications in abstract algebra
  • Explore the relationship between

    polynomial degree

    and field isomorphisms
  • Investigate the implications of

    modular arithmetic

    in polynomial equations

USEFUL FOR

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

rukawakaede
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Hi, could anyone solve my confusion?

Let [itex]p[/itex] be a prime and let [itex]x=\zeta_p[/itex] to be a primitive pth root of unity.

How could we conclude

[tex]\frac{\mathbf{Z}[x]/(1-x)}{(x^{p-1}+\cdots+x+1)}=\mathbf{Z}/p\mathbf{Z}=\mathbb{F}_p ?[/tex]

This should be obvious but it seems that I missed something. Could anyone help?Am I correct to infer that if we want to force [itex]1-x = 0[/itex], i.e. [itex]x = 1[/itex] and hence [itex]\mathbf{Z}[x]/(1-x)=\mathbf{Z}[/itex]. So [itex]\frac{\mathbf{Z}[x]/(1-x)}{(x^{p-1}+\cdots+x+1)}=Z/(p)[/itex] and therefore gives the result?
 
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Are you sure the problem reads Z[x]/(1-x)(x^{p-1}+...+x+1) = Z/pZ and
not Z[x]/(1-x)/(x^{p-2}+...+x+1) = Z/pZ ?
 
Eynstone said:
Are you sure the problem reads Z[x]/(1-x)(x^{p-1}+...+x+1) = Z/pZ and
not Z[x]/(1-x)/(x^{p-2}+...+x+1) = Z/pZ ?

I am pretty sure it is Z[x]/(1-x)(x^{p-1}+...+x+1) = Z/pZ

it would be good that you write out why you think it is not the case.
 

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