I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ...(adsbygoogle = window.adsbygoogle || []).push({});

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:

In the above text from Lovett we read the following:

" ... ... The element ##\sqrt[3]{2} \notin F## and ##\sqrt[3]{2}## has minimal polynomial ...

##m(t) = t^3 - x##.

However,

##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

##m(t) = t^3 - x##?

Indeed, what exactly is ##t##? ... what is ##x##? Which fields/rings do ##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

##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 ... ... ]

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

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