Irreducible polynomial on polynomial ring

In summary, proving that x^2+1 is irreducible in Z_p[x] requires finding roots in the ring, which would mean solving for x^2=-1 (mod p) or x^2+1=k(3+4m) for some k. Induction on m is not successful as x^2+1 is only reducible on Z_p[x] if p is prime, not necessarily for all m. A possible solution involves using Fermat's little theorem and a lemma stating that if a in a commutative ring with identity is invertible, then a^n=1 and a^m=1 implies a^gcd(n,m)=1.
  • #1
SN1987a
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How would I prove that [itex]x^2+1[/itex] is irreducible in [itex] Z_p[x][/itex], where p is an odd prime of the form 3+4m.

I know that for it to be rreducible, it has to have roots in the ring. So [itex]x^2=-1 (mod p)[/itex]. Or [itex] x^2+1=k(3+4m) [/itex], for some k. I tried induction on m, but it does not work because [itex}x^2+1[/itex] is only reducible on [itex] Z_p[x][/itex] if p is prime, which is not the case for all m. Apperently, there exists a two-line solution.

Any tips would be appreciated.
 
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  • #2
Hint: Fermat's little theorem and this lemma: if R is a commutative ring with identity, and a in R is invertible, then a^n=1 and a^m=1 => a^gcd(n,m)=1.
 

1. What is an irreducible polynomial?

An irreducible polynomial is a polynomial that cannot be factored into two or more polynomials with lower degrees. In other words, it cannot be broken down into simpler polynomial expressions.

2. What is the significance of an irreducible polynomial in a polynomial ring?

In a polynomial ring, an irreducible polynomial plays a similar role to a prime number in the ring of integers. It helps to identify unique elements and allows for more efficient calculations and simplification of expressions.

3. How is an irreducible polynomial identified?

There are various methods for identifying irreducible polynomials, such as the Eisenstein's criterion, the rational root test, and the reducibility test. These methods involve checking for specific properties of the polynomial, such as its coefficients and degree.

4. Can an irreducible polynomial exist in a polynomial ring with coefficients from a specific field?

Yes, an irreducible polynomial can exist in a polynomial ring with coefficients from any field. However, the process of identifying irreducible polynomials may differ depending on the field.

5. How are irreducible polynomials used in practical applications?

Irreducible polynomials have various applications in fields such as cryptography, coding theory, and signal processing. They are used to generate finite fields, construct error-correcting codes, and design filters and waveforms.

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