Separable Polynomials - Paul E Bland's definition and example ....

[Math Amateur]

I am reading Paul E Bland's book: The Basics of Abstract Algebra and I am trying to understand his definition of "separable polynomial" and his second example ...

Bland defines a separable polynomial as follows:https://www.physicsforums.com/attachments/6636... and Bland's second example is as follows:https://www.physicsforums.com/attachments/6637I am uncomfortable with, and do not fully understand this definition and am uncomfortable with the example as well ... I hope someone can clarify my difficulties and problems ...

The first and second sentences of the definition above seem to lead to different notions of separability to me ...

The first sentence of the definition:

" ... ... For a field \(\displaystyle F\), an irreducible polynomial \(\displaystyle f(x) \in F[x]\) of degree \(\displaystyle n\) is said to be separable if \(\displaystyle f(x)\) has \(\displaystyle n\) distinct roots in its splitting field ... ... "

Under this definition of a separable polynomial, the polynomial in Bland's example:

\(\displaystyle f(x) = (x^2 + 2)^2 (x^2 - 3)\)

is of degree \(\displaystyle n = 6\) and splits in (among other fields) in \(\displaystyle \mathbb{C}\) and does NOT have \(\displaystyle 6\) distinct roots (as the roots \(\displaystyle \pm \sqrt{2} i\) are repeated) ... ...(EDIT ... hmm ... but i guess you could argue that \(\displaystyle f(x)\) is not irreducible ... is that the key to my confusion ...?)... BUT ... The second sentence of Bland's definition reads:

" ... ... A polynomial in \(\displaystyle F[x]\) is said to be separable if each of its irreducible factors is separable if each of its irreducible factors is separable ... "

Well ... under this definition I (uncomfortably) go along with Bland's analysis of

\(\displaystyle f(x) = (x^2 + 2) (x^2 + 2) (x^2 - 3)\)

in his example ...BUT i remain uncomfortable with this ... I cannot think of an example of a case where this definition gives rise to a repeated root ...can someone please give a simple example of a polynomial with a repeated root under Bland's definition ...
Help will be much appreciated ...

Peter

 

[jvicens]

Hello Peter,

Thank you for reaching out for clarification on Bland's definition of separable polynomials. I can understand why you are feeling uncomfortable with the definition and the example provided.

To start, let's look at the first sentence of Bland's definition: "For a field F, an irreducible polynomial f(x) \in F[x] of degree n is said to be separable if f(x) has n distinct roots in its splitting field." This definition is referring to a specific polynomial, f(x), and its behavior in its splitting field. In other words, if we take the polynomial f(x) and factor it into irreducible polynomials in its splitting field, we should get n distinct factors. This means that the polynomial itself should not have any repeated roots in its splitting field.

Now, let's look at the second sentence of Bland's definition: "A polynomial in F[x] is said to be separable if each of its irreducible factors is separable." This definition is referring to a polynomial in general, not just a specific one like in the first sentence. This means that for any polynomial in F[x], if we factor it into irreducible polynomials, each of those irreducible factors should also be separable according to the first definition.

To answer your question about a polynomial with a repeated root under Bland's definition, consider the polynomial f(x) = (x+1)^3. This polynomial is separable according to the first definition because it has 3 distinct roots (x = -1, -1, -1) in its splitting field. However, when we factor it into irreducible polynomials, we get (x+1)^3 which is not separable according to the second definition because its only irreducible factor is not separable (it has a repeated root).

I hope this helps to clarify your confusion. It is important to note that there are different definitions of separable polynomials in abstract algebra, so it is possible that you have come across a different definition in your studies. If you are still feeling unsure, I would suggest consulting with your professor or a fellow mathematician for further clarification. Keep up the good work in your studies!