Primitive 5th root of unity extension

In summary, a primitive 5th root of unity extension is an extension field of the field of rational numbers, containing only the 5th roots of unity. It is different from other extensions and has significance in construction and solving equations. It is used in various mathematical fields and has real-life applications in signal processing, coding theory, and cryptography.
  • #1
mrbohn1
97
0
Hi,

Let E = Q(a), where a is a primitive fifth root of unity. Find a basis for E as a vector space over Q, and express a/a-3 in terms of this basis.

I can find a basis for E: {1, a, a^2, a^3}, but am not sure how to express a/a-3 in terms of this basis. Any help much appreciated.
 
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  • #2
Hint: If |z|<1, do you know of another way of writing 1/(1-z)?
 
Last edited:
  • #3
aaah...I see - thanks. I wasn't expecting to have to use analysis in a galois theory exercise...
 

FAQ: Primitive 5th root of unity extension

What is a primitive 5th root of unity extension?

A primitive 5th root of unity extension is an extension field of the field of rational numbers, defined as the smallest field containing all 5th roots of unity.

How is a primitive 5th root of unity extension different from other extensions?

A primitive 5th root of unity extension is different from other extensions because it contains only the 5th roots of unity, while other extensions may contain a larger set of elements.

What is the significance of the 5th root of unity in this extension?

The 5th root of unity is significant in this extension because it is the smallest number that, when raised to the power of 5, results in 1. This is important for constructing the extension field.

How is a primitive 5th root of unity extension used in mathematics?

A primitive 5th root of unity extension is used in various mathematical fields, such as algebra, number theory, and geometry, to solve equations and explore the properties of 5th roots of unity.

What are some real-life applications of a primitive 5th root of unity extension?

A primitive 5th root of unity extension has applications in signal processing, coding theory, and cryptography, where the concept of roots of unity is used to analyze and manipulate data or codes.

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