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a question for the irreducibility test for degree 2 or 3 polynomial

The ring $ℤ/nℤ$ is a field if and only if $n$ is prime'shifted' polynomial preserves validity of Eisenstein irreducibility criterion of original, in finite fields?About irreducibility of a particular class of bihomogeneous polynomialsProving irreducibility of polynomialIrreducibility of a polynomial with degree $4$question on testing irreducibility of a polynomialDetermine the irreducibility of polynomial.Is there a way to find one irreducible polynomial of degree n in the field Z2Mod $p$ irreducibility test proof questionFor every $n$, there exists a polynomial of degree $n$ with Galois Group $S_n$Why is $p_3(x) = 9x^2-3=3(xsqrt3-1)(xsqrt3+1)$ reducible over integers?

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I have a question for the irreducibility test for degree 2 or 3 polynomial. The test states: "Let F be a field, f(x) ∈ F[x], and degf(x) = 2 or 3. Then f(x) is reducible over F if and only if f(x) has a zero in F."

In the textbook, it states the test can be applied to polynomials in Z2[x] or Z3[x]. However, I don't think Z2 and Z3 are fields, just like Z is not field. Can anyone help? Thanks!

abstract-algebra polynomials irreducible-polynomials

share|cite|improve this question

edited 2 days ago

Sil

5,16421643

asked Mar 3 at 16:33

RandomThinkerRandomThinker

474

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  Wrong! Z2 and Z3 are fields, if this means $mathbb Z/2mathbb Z$, respectively $mathbb Z/3mathbb Z$.
  $endgroup$
  – user26857
  Mar 3 at 16:34
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  For a prime $p,$ the ring $mathbb Z / p mathbb Z$ is a field. The reason there are multiplicative inverses: if $n$ is an integer not divisible by $p,$ there are integers $x,y$ such that $nx-py = 1,$ or $nx= 1+py,$ finally $nx equiv 1 pmod p$
  $endgroup$
  – Will Jagy
  Mar 3 at 16:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I see! Thanks guys!
  $endgroup$
  – RandomThinker
  Mar 3 at 16:49
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See this duplicate about $Bbb Z/n$.
  $endgroup$
  – Dietrich Burde
  Mar 3 at 17:49
  

  
  
  
  

add a comment | 

0

$begingroup$

I have a question for the irreducibility test for degree 2 or 3 polynomial. The test states: "Let F be a field, f(x) ∈ F[x], and degf(x) = 2 or 3. Then f(x) is reducible over F if and only if f(x) has a zero in F."

In the textbook, it states the test can be applied to polynomials in Z2[x] or Z3[x]. However, I don't think Z2 and Z3 are fields, just like Z is not field. Can anyone help? Thanks!

abstract-algebra polynomials irreducible-polynomials

share|cite|improve this question

edited 2 days ago

Sil

5,16421643

asked Mar 3 at 16:33

RandomThinkerRandomThinker

474

$endgroup$

• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  Wrong! Z2 and Z3 are fields, if this means $mathbb Z/2mathbb Z$, respectively $mathbb Z/3mathbb Z$.
  $endgroup$
  – user26857
  Mar 3 at 16:34
  
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  For a prime $p,$ the ring $mathbb Z / p mathbb Z$ is a field. The reason there are multiplicative inverses: if $n$ is an integer not divisible by $p,$ there are integers $x,y$ such that $nx-py = 1,$ or $nx= 1+py,$ finally $nx equiv 1 pmod p$
  $endgroup$
  – Will Jagy
  Mar 3 at 16:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I see! Thanks guys!
  $endgroup$
  – RandomThinker
  Mar 3 at 16:49
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  See this duplicate about $Bbb Z/n$.
  $endgroup$
  – Dietrich Burde
  Mar 3 at 17:49
  

  
  
  
  

add a comment | 

$begingroup$

I have a question for the irreducibility test for degree 2 or 3 polynomial. The test states: "Let F be a field, f(x) ∈ F[x], and degf(x) = 2 or 3. Then f(x) is reducible over F if and only if f(x) has a zero in F."

In the textbook, it states the test can be applied to polynomials in Z2[x] or Z3[x]. However, I don't think Z2 and Z3 are fields, just like Z is not field. Can anyone help? Thanks!

abstract-algebra polynomials irreducible-polynomials

share|cite|improve this question

edited 2 days ago

Sil

5,16421643

asked Mar 3 at 16:33

RandomThinkerRandomThinker

474

$endgroup$

share|cite|improve this question

edited 2 days ago

Sil

5,16421643

asked Mar 3 at 16:33

RandomThinkerRandomThinker

474

share|cite|improve this question

edited 2 days ago

Sil

5,16421643

asked Mar 3 at 16:33

RandomThinkerRandomThinker

474

edited 2 days ago

Sil

5,16421643

edited 2 days ago

Sil

5,16421643

asked Mar 3 at 16:33

RandomThinkerRandomThinker

474

asked Mar 3 at 16:33

RandomThinkerRandomThinker

474