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 ago

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