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 ... ... ]

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads - Example Inseparable Polynomial | Date |
---|---|

A Example of how a rotation matrix preserves symmetry of PDE | Feb 10, 2018 |

I Projective Representations: a simple example | Jan 17, 2018 |

B Galois Groups ... A&F Example 47.7 ... ... | Jul 7, 2017 |

I Finite Extensions - A&F Example 44.2 ... ... | Jul 3, 2017 |

A Galois theorem in general algebraic extensions | Apr 29, 2017 |

**Physics Forums - The Fusion of Science and Community**