Factorization of Polynomials - Irreducibles - Anderson and Feil

[Math Amateur]

I am reading Anderson and Feil - A First Course in Abstract Algebra.

On page 56 (see attached) ANderson and Feil show that the polynomial $$f = x^2 + 2$$ is irreducible in $$\mathbb{Q} [x]$$

After this they challenge the reader with the following exercise:

Show that $$x^4 + 2$$ is irreducible in $$\mathbb{Q} [x]$$. taking your lead from the discussion of $$x^2 + 2$$ above. (see attached)

Can anyone help me to show this in the manner requested. Would appreciate the help.

Peter

 

[Sudharaka]

I am reading Anderson and Feil - A First Course in Abstract Algebra.

On page 56 (see attached) ANderson and Feil show that the polynomial $$f = x^2 + 2$$ is irreducible in $$\mathbb{Q} [x]$$

After this they challenge the reader with the following exercise:

Show that $$x^4 + 2$$ is irreducible in $$\mathbb{Q} [x]$$. taking your lead from the discussion of $$x^2 + 2$$ above. (see attached)

Can anyone help me to show this in the manner requested. Would appreciate the help.

Peter

Hi Peter, :)

I don't see any attachments in your post. You can use a image hosting website such as TinyPic to upload images and link them here, if you have trouble attaching files.

To show that \(x^4 + 2\) is irreducible over \(\mathbb{Q}[x]\) you can use Eisenstein's Irreducibility Criterion.

 

[Math Amateur]

Hi Peter, :)

I don't see any attachments in your post. You can use a image hosting website such as TinyPic to upload images and link them here, if you have trouble attaching files.

To show that \(x^4 + 2\) is irreducible over \(\mathbb{Q}[x]\) you can use Eisenstein's Irreducibility Criterion.

Thanks - most helpful - appreciate your help

The reason I did not upload the attachement was that I could not delete my old attachements - about 5 or so are there and they exceed my allowed quota _ I cannot seem to delete them

Peter

 

[Sudharaka]

Thanks - most helpful - appreciate your help

The reason I did not upload the attachement was that I could not delete my old attachements - about 5 or so are there and they exceed my allowed quota _ I cannot seem to delete them

Peter

To if you want to delete your previous attachments go to "http://www.mathhelpboards.com/usercp.php" and then click on the "http://www.mathhelpboards.com/profile.php?do=editattachments" under the "My Settings" pane. Hope this will work for you. :)

 

[blue_raver22]

I am not an expert in abstract algebra but I can provide some insights on factorization of polynomials and irreducibles based on my understanding.

Factorization of polynomials is the process of breaking down a polynomial into a product of simpler polynomials. This is similar to how we break down numbers into prime factors. In abstract algebra, the concept of irreducibles is important as it helps us identify polynomials that cannot be factored any further.

In the given exercise, we are asked to show that x^4 + 2 is irreducible in \mathbb{Q}[x]. To do this, we can follow the approach used for x^2 + 2. First, we assume that x^4 + 2 can be factored into two polynomials, say f(x) and g(x), both with coefficients in \mathbb{Q}. This would mean that x^4 + 2 = f(x)g(x).

Next, we can use the fact that x^2 + 2 is irreducible in \mathbb{Q}[x]. This implies that x^2 + 2 cannot be factored into two polynomials with coefficients in \mathbb{Q}. Therefore, we can assume that neither f(x) nor g(x) is equal to x^2 + 2.

Now, we can consider the coefficients of x^4 + 2. Since the coefficients are in \mathbb{Q}, we can write them as fractions in lowest terms. Let's say the coefficient of x^4 is a/b, where a and b are integers with no common factors. Similarly, the constant term is c/d where c and d are integers with no common factors.

Using the assumption that x^4 + 2 = f(x)g(x), we can write out the coefficients of f(x) and g(x) as fractions in lowest terms. This would give us the following equations:

f(x) = (ax^2 + e)/(bx^2 + g)
g(x) = (cx^2 + h)/(dx^2 + i)

Where e, g, h and i are integers with no common factors.

Now, we can substitute these expressions into the equation x^4 + 2 = f(x)g(x) and equate the coefficients of x^2 and the constant term. This would give us the following equations:

a/b + c/d = 0