Proving Irreducibility of x^4 −7 Using Polynomial Theorems

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Thread starter Poirot1
Start date Mar 11, 2012
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Polynomial

In summary, polynomial theorems are helpful in determining the irreducibility of a polynomial. The first step in using these theorems is to factor the polynomial into its irreducible components. These theorems can be applied to polynomials of any degree, but may become more complex with higher degrees. Some commonly used polynomial theorems include the rational root theorem, the Eisenstein criterion, and irreducibility criteria for quadratic and cubic polynomials. It is possible for a polynomial to be irreducible according to one theorem but reducible according to another, highlighting the importance of using multiple theorems for proving irreducibility.

Mar 11, 2012

Poirot1

Explain why the polynomial x^4 −7 is irreducible over Q, quoting any theorems you use.

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Mar 11, 2012

tarantella

Which theorems related to irreducibility do you know?

Related to Proving Irreducibility of x^4 −7 Using Polynomial Theorems

1. How do polynomial theorems help in proving irreducibility?

Polynomial theorems provide a set of rules and criteria for determining whether a polynomial is irreducible. By applying these theorems to the polynomial x^4 −7, we can determine if it is irreducible or not.

2. What is the first step in using polynomial theorems to prove irreducibility?

The first step is to factor the polynomial into its irreducible components. For x^4 −7, this would be (x^2+√7)(x^2-√7).

3. Can polynomial theorems be used for any degree of polynomial?

Yes, polynomial theorems can be used for polynomials of any degree. However, the complexity of the theorems may increase with higher degrees.

4. What are some commonly used polynomial theorems for proving irreducibility?

Some commonly used polynomial theorems include the rational root theorem, the Eisenstein criterion, and the irreducibility criteria for quadratic and cubic polynomials.

5. Is it possible for a polynomial to be irreducible using one theorem but reducible using another?

Yes, it is possible for a polynomial to satisfy the criteria of one theorem but not another. This is why it is important to use multiple theorems to determine the irreducibility of a polynomial.