Criterion for Irreducibility of a polynomial in several variables?

In summary, the criterion for the irreducibility of a polynomial in several variables over an algebraically closed field is that if some specialization of the variables gives an irreducible polynomial in one variable, then the multivariate polynomial is also irreducible. This can be shown by finding elements in the field such that substituting them for the variables (except the first one) results in an irreducible polynomial in the first variable. However, this may not always work, as seen in the example of x^2y+y over the reals, where it is irreducible as a polynomial in x but reducible as a polynomial in x and y.
  • #1
GargleBlast42
28
0
Is there any criterion for the irreducibility of a polynomial in several variables over an algebraically closed field (or specifically for the complex numbers)? For one variable, we know this is simply that only degree one polynomials are irreducible, is there anything similar for several variables?
 
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  • #2
If you can show that some specialization of the variables gives an irreducible polynomial in one variable, then this implies that the multivariate polynomial is irreducible.

So suppose you have some polynomial in K[x_1,x_2,x_3,...], where K is a field, and the x_i are indeterminates. Then all you need to do is find elements: a_2, a_3, a_4,... in K such that substituting a_i for x_i (i>1) gives an irreducible polynomial in K[x_1].
 
  • #3
Also, every polynomial in x and y over C is a polynomial in x over C(y).
 
  • #4
mrbohn1 said:
If you can show that some specialization of the variables gives an irreducible polynomial in one variable, then this implies that the multivariate polynomial is irreducible.

So suppose you have some polynomial in K[x_1,x_2,x_3,...], where K is a field, and the x_i are indeterminates. Then all you need to do is find elements: a_2, a_3, a_4,... in K such that substituting a_i for x_i (i>1) gives an irreducible polynomial in K[x_1].

I'm not really sure if this works. Take for instance [tex]x^2 y+y[/tex] over the reals. As a polynomial in x, it is irreducible for any y, but as a polynomial in x,y it is obviously reducible [tex]x^2 y+y= y(x^2+1)[/tex]
 
  • #5
GargleBlast42 said:
As a polynomial in x, it is irreducible for any y
I do believe the zero polynomial counts as reducible.

edit: true but irrelevant
 
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  • #6
I don't think that the constant polynomial does count. I should have been clearer: this only works if you consider your multivariate polynomial as a polynomial in one variable over the field of functions in the other variables. In your example, the field would be R(y), and reducible in this case would refer only to the x variable.
 

1. What is the criterion for irreducibility of a polynomial in several variables?

The criterion for irreducibility of a polynomial in several variables is the Eisenstein's criterion, which states that a polynomial is irreducible if it satisfies the following conditions:

  • The leading coefficient is prime.
  • All other coefficients are divisible by the prime except the constant term.
  • The constant term is not divisible by the square of the prime.

2. How does the Eisenstein's criterion work?

The Eisenstein's criterion works by identifying a prime number that satisfies the conditions mentioned above. If such a prime exists, then the polynomial is irreducible over the field of rational numbers. This method can be used to quickly determine the irreducibility of polynomials without resorting to more complex methods.

3. Can the Eisenstein's criterion be used for polynomials with more than one variable?

Yes, the Eisenstein's criterion can be extended to polynomials with more than one variable. The conditions remain the same, except that the leading coefficient and constant term must now be prime polynomials. If these conditions are satisfied, then the polynomial is irreducible over the field of rational numbers.

4. Is the Eisenstein's criterion the only criterion for irreducibility of polynomials in several variables?

No, there are other criteria for determining the irreducibility of polynomials in several variables, such as the Newton's polygon method, the Jacobian criterion, and the Hilbert's irreducibility theorem. These methods may be more complex and require more calculations, but they can be useful when the Eisenstein's criterion cannot be applied.

5. Can the Eisenstein's criterion be used for all polynomials in several variables?

No, the Eisenstein's criterion can only be applied to polynomials with coefficients in the field of rational numbers. It cannot be used for polynomials with coefficients in other fields, such as integers or real numbers. In such cases, other methods must be used to determine the irreducibility of the polynomial.

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