how to find the Galois group?

[logistic_guy]

i really appreciate your standing with me so far blamocur

\(\displaystyle \sigma_2 + \sigma_3 + \sigma_5 = (a - b\sqrt{2} - c\sqrt{3} - d\sqrt{5})\) + \(\displaystyle (a + b\sqrt{2} + c\sqrt{3} - d\sqrt{5})\) + \(\displaystyle (a - b\sqrt{2} - c\sqrt{3} + d\sqrt{5})\) = \(\displaystyle a + a + a - b\sqrt{2} - c\sqrt{3} - d\sqrt{5}\) = \(\displaystyle \sigma_2\)

there is \(\displaystyle 8\) automorphisims in the list. i don't understand why do you chose only \(\displaystyle \sigma_3,\sigma_4,\sigma_6\). i don't see how these used to get others!

 

[blamocur]

i really appreciate your standing with me so far blamocur

\(\displaystyle \sigma_2 + \sigma_3 + \sigma_5 = (a - b\sqrt{2} - c\sqrt{3} - d\sqrt{5})\) + \(\displaystyle (a + b\sqrt{2} + c\sqrt{3} - d\sqrt{5})\) + \(\displaystyle (a - b\sqrt{2} - c\sqrt{3} + d\sqrt{5})\) = \(\displaystyle a + a + a - b\sqrt{2} - c\sqrt{3} - d\sqrt{5}\) = \(\displaystyle \sigma_2\)

there is \(\displaystyle 8\) automorphisims in the list. i don't understand why do you chose only \(\displaystyle \sigma_3,\sigma_4,\sigma_6\). i don't see how these used to get others!

I believe that all other automorphisms can be expressed as products/compositions of some or all of those three (https://en.wikipedia.org/wiki/Generating_set_of_a_group).

The expression you posted is not related to the group of automorphisms. Do you know what is the operation which defines that group?

 

[logistic_guy]

i read the page you send, but i don't see how this will help me to produce all automorphisims from \(\displaystyle \sigma_3\), \(\displaystyle \sigma_4\), \(\displaystyle \sigma_6\)

 

[blamocur]

What is the product of [imath]a_3[/imath] and [imath]a_4[/imath] ?

 

[logistic_guy]

i'm afraid normal multiplication don't work here. am i right?

\(\displaystyle \alpha_3 \alpha_4 \neq \left(a + b\sqrt{2} + c\sqrt{3} - d\sqrt{5}\right)\left(a + b\sqrt{2} - c\sqrt{3} + d\sqrt{5}\right)\)

if this invalid then i don't know the product of \(\displaystyle \alpha_3 \alpha_4\)

 

[blamocur]

i'm afraid normal multiplication don't work here. am i right?

\(\displaystyle \alpha_3 \alpha_4 \neq \left(a + b\sqrt{2} + c\sqrt{3} - d\sqrt{5}\right)\left(a + b\sqrt{2} - c\sqrt{3} + d\sqrt{5}\right)\)

if this invalid then i don't know the product of \(\displaystyle \alpha_3 \alpha_4\)

Seems that you are trying to guess. This thread will last forever if you use it instead of learning the basic definitions and properties first. You cannot solve problems with Galois groups if you don't understand automorphisms and how they are multiplied.

 

[logistic_guy]

Seems that you are trying to guess. This thread will last forever if you use it instead of learning the basic definitions and properties first. You cannot solve problems with Galois groups if you don't understand automorphisms and how they are multiplied.

this mean i should stop this thread for awhile to get more knowledge in the fundamental definitions. thank blamocur.