Question
In: Advanced Math
Definition Call a number eld super-constructible if there is a tower of eld extensions, =0 1...
Definition Call a number eld super-constructible if there is a tower of eld extensions,
=0 1 =, where[+1 ]=1,2,or3.
1) Give an example of a eld that is not super-constructible. Prove that your example is correct.
2) Give an example of a super-constructible eld that is not a constructible number eld. Prove that your answer is correct.
Solutions
Expert Solution
a) Let 
be the splitting field of 
. We know that
and
Thus,
and its only subgroups are
By fundamental theorem of Galois theory, the intermediate fields are as follows:
where 
and 
. Hence. in particular, there can be no tower
having degree of each extensions.
.
Thus, 
is not super-constructible.
b) Let
be the splitting field of 
. We know that
and
Thus,
and its only subgroups are
By fundamental theorem of Galois theory, the intermediate fields are as follows:
where 
and
. Hence. in particular, the tower
is such that degree of each extension satisfies.
Thus,
is super-constructible. However, note that it is not constructible
because
is not an integral power of 
.
Colby Messinger answered 1 year agoRelated Solutions
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