Irreducibility of Polynomial part deux

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In summary, the conversation discusses a theorem known as Gauss' Lemma, which states that if a polynomial in Z[x] can be factored into two polynomials in Q[x], then the coefficients can also be selected to be in Z[x]. The conversation also mentions a method of changing the coefficients of a polynomial into elements of a prime congruence class, and how this can show that the original polynomial is irreducible. The conversation ends with a recommendation to look at some free algebra notes on the speaker's website.
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Square1
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Ok I promise this time it is not a homework type question.

If someone could direct with a name of a theorem here, then I'll go ahead and google it, otherwise have a look. I have a chunk of notes that I'm confused about. We're not shown any proofs here or any explanations, just what is seen. It was talking about in somewhat of a "mention in passing" way so maybe I am not supposed to look too deep into it and just accept the results. Here it is...Basically we are told that if we have a polynomial f in Z[x], if f = gh where g and h belong t Q[x], it can be shown that the coefficients can be selected to be in Z[x]. We told this is Gauss' Lemma. OK. Then it goes, we change the given polynomial into the corresponding polynomial that has the coefficients changed into elements of a prime congruence class. g and h are defined the same, then "f-bar" = "g-bar" times "h-bar". If p prime is chosen right, it shows that if "f-bar" is irreducible, then f is irreducible.

I'll start by asking does anyone know what this is describing, and if we have a name for it?
 
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Ok just found it actually. Case closed -.-
 
  • #3
you might look at some of my free algebra notes on my website. see my public profile for my web address under contact info. or just click on my name at left and the pulldown menu gives a link to it.
 
  • #4
Thanks for the suggestion, roy :) . I'll keep this mind.
 
  • #5


This is describing the concept of irreducibility of polynomials over a ring, specifically in the context of Gauss' Lemma and the use of prime congruence classes. This theorem is known as the "Eisenstein's Irreducibility Criterion" and states that if a polynomial with integer coefficients can be written as a product of two polynomials with rational coefficients, then it can also be written as a product of two polynomials with integer coefficients, as long as a specific prime number is chosen. This criterion is useful in determining the irreducibility of a polynomial over a given ring.
 

1. What is the concept of Irreducibility of Polynomial part deux?

The concept of Irreducibility of Polynomial part deux refers to the property of a polynomial where it cannot be factored into polynomials of lower degree with coefficients in the same field. In simpler terms, it means that the polynomial cannot be broken down into simpler components.

2. Why is the Irreducibility of Polynomial part deux important in mathematics?

The concept of Irreducibility of Polynomial part deux is important in mathematics because it helps us understand the structure and behavior of polynomials. It also has applications in various mathematical fields such as algebra, geometry, and number theory.

3. How can we determine if a polynomial is irreducible?

To determine if a polynomial is irreducible, we can use various methods such as the Rational Root Theorem, Eisenstein's Criterion, and the Reduction Modulo p method. These methods help us identify any possible factors of the polynomial and determine if it can be reduced further.

4. Can a polynomial be both irreducible and reducible?

No, a polynomial cannot be both irreducible and reducible. By definition, a polynomial is either irreducible or reducible. If a polynomial cannot be factored into polynomials of lower degree, it is considered irreducible. On the other hand, if it can be factored, it is considered reducible.

5. What is the difference between the Irreducibility of Polynomial part deux and part one?

The main difference between Irreducibility of Polynomial part deux and part one is that part deux considers polynomials with coefficients in the same field, while part one considers polynomials with coefficients in a different field. This means that a polynomial can be irreducible in one field but reducible in another field.

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