Number theory Extensions

In summary, the conversation discusses a question about finding the minimum polynomial of a given example and how to interpret the notation used. The original question asks for clarification on the notation L=F(a) and how it relates to degree of the minimum polynomial. The conversation then moves on to discussing an example found online and clarifying a step in the solution. The final question is about choosing between two different polynomials with the same degree and whether it matters which one is chosen. One person explains that it does not matter and provides a correction for the incorrect step in the calculation.
  • #1
Firepanda
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Wouldn't mind a hint on how to start part iii), thanks.

edit: in my notes i have for a similar question:

'L=Q(20.5, 30.5)
F=Q(60.5)

degree of the min polynomial = 2, because L=F(a) and [L:F]=2
' (a = alpha)

Could someone clarify what L=F(a) means so I can understand the example? Thanks
 
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  • #2
I have found an example of how to do it here:

http://www.math.niu.edu/~beachy/abstract_algebra/study_guide/soln6.html

Though, can someone explain to me on their example of the last line to the solution of 2.b) as to why they chose

u -i = 20.5

and not

u - 20.5 = i

or does it not matter as both give you a poly of degree 4?
 
  • #3
The answer to the question in your second post is that you may use either expression for the reason you gave ("... both give you a poly of degree 4"). In fact, they give you the same poly. Does that clear up your original question?

Petek
 
  • #4
Petek said:
The answer to the question in your second post is that you may use either expression for the reason you gave ("... both give you a poly of degree 4"). In fact, they give you the same poly. Does that clear up your original question?

Petek

Do they give the same poly?

If I do it that way I get

u - 20.5 = i

u2 - 2u20.5 - 2 = -1

u2 - 1 = 2u20.5

u4 - 2u2 + 1 = 8u2

u4 - 10u2 + 1 = 0

and their answer was

u4 - 2u2 + 9 = 0


Hence I have two different polynomials for the question I'm trying to do, both degree 4 and monic and I'm not sure which to choose.
 
  • #5
In your calculation, this step

u2 - 2u20.5 - 2 = -1

is incorrect. Do you see why?

Petek
 

1. What is number theory and what are its extensions?

Number theory is a branch of mathematics that deals with the properties and relationships of numbers. It explores patterns and structures in numbers and their properties such as divisibility, prime numbers, and integer solutions. Number theory extensions refer to the further developments and applications of number theory concepts and techniques.

2. How are number theory extensions used in cryptography?

Number theory extensions, particularly in the field of algebraic number theory, are used in cryptography to develop secure encryption and decryption algorithms. These extensions provide a mathematical basis for the creation of public and private key pairs, which are used to encrypt and decrypt messages.

3. What are some real-world applications of number theory extensions?

Number theory extensions have various real-world applications, such as in coding theory, which is used in communication systems to detect and correct errors in transmitted data. They are also used in the development of efficient algorithms for prime factorization, which is important in cryptography and computer security.

4. Can you explain the concept of modular arithmetic in number theory extensions?

Modular arithmetic is an extension of number theory that deals with operations on remainders. It involves dividing a number by a modulus and finding the remainder, which is then used to perform arithmetic operations. Modular arithmetic has various applications, including in computer science for generating random numbers and in cryptography for creating secure hash functions.

5. How do number theory extensions relate to other areas of mathematics?

Number theory extensions have connections with many other areas of mathematics, such as algebra, geometry, and analysis. For example, algebraic number theory studies the properties of algebraic numbers, which are solutions to polynomial equations with integer coefficients. Analytic number theory uses tools from calculus and analysis to study the distribution of prime numbers. Number theory also has links to combinatorics, graph theory, and even physics.

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