# Extending a field by a 16th primitive root of unity

• PsychonautQQ
In summary: The latter is always 16, which is why ##c## is a primitive 16-th root of unity.In summary, the only subfield M of Q(c) such that [M:Q] = 2 is Q(c^8). Other powers of c such as c^2, c^4, c^6, c^10, and c^12 do not generate extension fields over Q with a degree of 2.
PsychonautQQ

## Homework Statement

let c be a primitive 16th root of unity. How many subfields M<Q(c) are there such that [M:Q] = 2

## The Attempt at a Solution

I think the only subfield M of Q(c) such that [M:Q] = 2 is Q(c^8). Then M = {a+b(c^8) such that a,b are elements of Q}. I'm thinking about the other powers of c and trying to think if any other would generate an extension field over Q with a degree of 2. Any number that's relatively prime to 16 would be another primitive 16th root of unity, so we can throw out all odd numbers. Q(c^2) and Q(c^14) would both be degree 8, Q(c^4) and Q(c^12) would both be degree four, Q(c^6) would be of degree 8 and Q(c^10) would also be of degree 8. So it's only [Q(c^8):Q]= 2 correct?

PsychonautQQ said:
I think the only subfield M of Q(c) such that [M:Q] = 2 is Q(c^8).
Think again about that '8' exponent. Note that ##c^8=-1##, which is in ##\mathbb Q##, so ##\mathbb Q(c^8)=\mathbb Q## (ie the extension is trivial), whence ##[\mathbb Q(c^8):\mathbb Q]=1##, not 2.
PsychonautQQ said:
Q(c^4) and Q(c^12) would both be degree four
Why? I think you are confusing the degree of the extension ##[\mathbb Q(c^k):\mathbb Q]##, which is the dimension of the vector space, with the order of the element ##c^k## in the multiplicative group ##\{c^j\ :\ j\in\{0,1,...,15\}\}##.

## What is a 16th primitive root of unity?

A 16th primitive root of unity is a complex number that, when raised to the power of 16, equals 1. It is also known as a 16th root of unity, and is one of the solutions to the equation x^(16) = 1.

## Why is it important to extend a field by a 16th primitive root of unity?

Extending a field by a 16th primitive root of unity allows for more complex mathematical operations to be performed, specifically in the field of abstract algebra. It also has applications in areas such as cryptography and signal processing.

## How is a field extended by a 16th primitive root of unity?

A field can be extended by a 16th primitive root of unity by adding the root to the existing field, creating a larger field with more elements. This process is known as a field extension and is a common technique in abstract algebra.

## What are the properties of a 16th primitive root of unity?

A 16th primitive root of unity has the property that when it is raised to the power of any positive integer less than 16, it will not equal 1. It is also a solution to the equation x^(16) = 1, and is a complex number with a magnitude of 1.

## Can a field be extended by a non-primitive root of unity?

Yes, a field can be extended by a non-primitive root of unity. However, the resulting field will not have the same properties as a field extended by a primitive root of unity, such as the ability to perform all complex mathematical operations.

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