Example of an Inseparable Polynomial .... Lovett, Page 371 ....

[Math Amateur]

I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ...

I am currently focused on Chapter 7: Field Extensions ... ...

I need help with Example 7.7.4 on page 371 ...Example 7.7.4 reads as follows:
View attachment 6651
In the above text from Lovett we read the following:" ... ... The element \(\displaystyle \sqrt[3]{2} \notin F\) and \(\displaystyle \sqrt[3]{2}\) has minimal polynomial m\(\displaystyle m(t) = t^3 - x\).However,\(\displaystyle m(t) = t^3 - 3t^2 \sqrt[3]{2} + 3t x^{ 2/3 } - x = (t - \sqrt[3]{2} )^3 \)... ... ... ... "
My questions are as follows:
Question 1

How does Lovett establish that the minimum polynomial is

\(\displaystyle m(t) = t^3 - x\)?Indeed, what exactly is \(\displaystyle t\)? ... what is \(\displaystyle x\)? Which fields/rings do \(\displaystyle t,x\) belong to?[My apologies for asking basic questions ... but unsure of the nature of this example!]Question 2


How does Lovett establish that \(\displaystyle m(t) = t^3 - 3t^2 \sqrt[3]{2} + 3t x^{ 2/3 } - x = (t - \sqrt[3]{2} )^3\)
Help will be appreciated ...

Peter
[NOTE: I understand that the issues in this example are similar to those of other of my posts ... but ... for clarity and to avoid mixing/confusing conversational threads and issues I have decided to post this example separately ... ... ]

 

[jvicens]

Hello Peter,

Thank you for reaching out for help with Example 7.7.4 from "Abstract Algebra: Structures and Applications" by Stephen Lovett. I am happy to assist you in understanding this example better.

Firstly, let's define the terms used in this example. In this context, t and x are both variables, and they do not belong to any specific field or ring. They are simply used as symbols to represent elements in a field extension. The field extension is denoted by F(\sqrt[3]{2}), which means that we are extending the field F by adding the element \sqrt[3]{2} to it.

Now, let's move on to your first question. Lovett establishes that the minimum polynomial for \sqrt[3]{2} is m(t) = t^3 - x by using the definition of a minimum polynomial. The minimum polynomial for an element \alpha in a field extension F(\alpha) is the smallest degree monic polynomial in F[t] (the ring of polynomials with coefficients in F) such that \alpha satisfies this polynomial. In simpler terms, the minimum polynomial is the smallest polynomial that has \alpha as a root. In this example, \sqrt[3]{2} is the root of the polynomial t^3 - x, and it is the smallest degree monic polynomial with \sqrt[3]{2} as a root. Therefore, it is the minimum polynomial for \sqrt[3]{2} in the field extension F(\sqrt[3]{2}).

Moving on to your second question, Lovett establishes that m(t) = t^3 - 3t^2 \sqrt[3]{2} + 3t x^{ 2/3 } - x = (t - \sqrt[3]{2} )^3 by using polynomial division. He divides t^3 - 3t^2 \sqrt[3]{2} + 3t x^{ 2/3 } - x by t - \sqrt[3]{2} to get the remainder of 0. This means that (t - \sqrt[3]{2} ) is a factor of t^3 - 3t^2 \sqrt[3]{2} + 3t x^{ 2/3 } - x. In other words, (t - \sqrt[3]{2} )^3 divides t^3 - 3