# Abstract Algebra: Extension Fields & Complex nth Roots

• Doom of Doom
In summary, the conversation discusses two problems from abstract algebra class related to algebraic closures and complex roots of unity. The first problem asks if K is the algebraic closure of E given certain conditions, while the second problem asks for a formula for the degree of a complex nth root of unity over the rational numbers. The conversation also suggests a possible formula for the second problem involving Euler's totient function.
Doom of Doom
Two problems from my abstract algebra class...

1)
Let K be the algebraic closure of a fi eld F and suppose E is a field such that  F $$F \subseteq E \subseteq K$$. Then is K the algebraic closure of E?

2)
Let $$n$$ be a natural number with $$n\geq2$$, and suppose that $$\omega$$ is a complex nth root on unity. Is there a formula for $$\left[\mathbb{Q}(\omega) : \mathbb{Q}\right]$$ ?

__________________

To 1), I must be missing something really silly, because it seems to me like it is obviously the case that K is also the algebraic closure of E, and that the proof should be easy. But I simply can't think of anything.

To 2) I would say no, but I am not exactly sure that I understand the question. For example, if n=8, then $$e^{i2\pi/8}$$ is a complex 8th root of unity such that $$\left[\mathbb{Q}(e^{i2\pi/8}) : \mathbb{Q}\right]=4$$. However, $$i$$ is also an 8th root of unity, but $$\left[\mathbb{Q}(i) : \mathbb{Q}\right]=2$$. Thus, for a given n, there is not necessarily a formula. Does this sound right?

1) Well for K to be the algebraic closure of E you must show:
a) K/E is algebraic.
b) If $g(x) \in E[x]$, then g(x) splits completely in K. (HINT: Remember that K is an algebraic closure of itself so $h(x) \in K[x]$ imply that h(x) splits completely in K).

2) Well $\omega$ is a specific nth root of unity so it's acceptable for your formula to behave differently when given $e^{i2\pi/8}$ and when given $e^{i\pi/8}$. You should probably look for a formula of the form:
$$\left[\mathbb{Q}\left(e^{ik\pi/n}\right) \, : \, \mathbb{Q} \right] = f(n,k)$$
so the formula can depend on both n and k, not just n.

Ok, for 2), I've got the formula

$$\left[\mathbb{Q}\left(e^{ik\pi/n}\right) \, : \, \mathbb{Q} \right] = \phi\left(\frac{n}{\gcd(n,k)}\right)$$,

where phi is Euler's totient funciton.

Is that right?

Now I just have to prove it...

## What is abstract algebra?

Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. It focuses on the properties and relationships of these structures, rather than specific numerical calculations.

## What are extension fields?

Extension fields are fields that contain elements from a smaller field, called the base field. They are created by adjoining new elements to the base field, resulting in a larger field with new properties and relationships.

## What are complex nth roots?

Complex nth roots are solutions to the equation x^n = a, where a is a complex number and n is a positive integer. They can be represented in the complex plane as points that are equidistant from the origin and form a regular polygon with n sides.

## What is the connection between extension fields and complex nth roots?

Extension fields are often used to construct fields containing the complex nth roots of a number. For example, the field of complex numbers is an extension of the field of real numbers and contains all the complex nth roots of unity.

## How are extension fields and complex nth roots used in real-world applications?

Extension fields and complex nth roots have many applications in mathematics, physics, and engineering. They are used in cryptography, coding theory, and signal processing, as well as in the study of symmetry and geometry.

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