Finding Infinite Units in \mathbb{Q}(\sqrt{21}) Using Continued Fractions

In summary, to find an infinite number of units in the ring of integers of the number field \mathbb{Q}(\sqrt{21}), one can use the continued fraction expansion of \sqrt{21} and the solutions to Pell's equation x^2 - 21y^2 = 1. This allows for the derivation of a fundamental unit that can be raised to any power and still remain a unit. Additional details and examples can be found in textbooks and further advice can be given for specific questions.
  • #1
math_grl
49
0
How do you find an infinite number of units of [tex]\mathbb{Q}(\sqrt{21})[/tex] using the [tex]\sqrt(21)[/tex]? I saw one example using continued fractions but do not know how to apply it in this case. I do have the periodic form of the continued fraction of [tex]\sqrt(21)[/tex].
 
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  • #2
Q(sqrt(21)) is a field, every element except 0 is a unit. a/(a^2+b^2)-b*sqrt(21)/(a^2+b^2) is an inverse to a+b*sqrt(21) for rational a and b.
 
  • #3
I interpret the original question to be "Find the units in the ring of integers of the number field [itex]\mathbb{Q}(\sqrt{21})[/itex]. If so, then you may know that the units in that ring correspond to solutions of the so-called Pell's equation [itex]x^2 - 21y^2 = 1[/itex]. Furthermore, solutions to Pell's equation can be obtained from the continued fraction expansion of [itex]\sqrt{21}[/itex].

If this is what you're looking for, then your textbook probably has more details and examples. I'd be glad to give more advice if you have specific questions.
 
  • #4
Yes Petek you hit the nail on the head, actually the book I'm reading seems to have no details and the few resources I looked up online just said we can derive some particular unit that if you raise it to the nth power, it's still a unit, from the pell's equation. I know the fundamental unit for [tex]\mathbb{Q}(\sqrt(21))[/tex] I'm just really unclear on the procedure or computation it involves to get there or to at least find a unit such as the one I described above.
 
  • #5
Thanks to Petek I was able to clear up my misunderstanding.
 

1. What is an infinite number of units?

An infinite number of units is a theoretical concept in mathematics and physics where a quantity is said to be unlimited or boundless, with no specific or defined value.

2. Is an infinite number of units larger than any other number?

Yes, an infinite number of units is considered to be larger than any other number. This is because it represents a quantity that is limitless and cannot be reached or calculated.

3. How is an infinite number of units used in scientific calculations?

An infinite number of units is often used in theoretical calculations and models, as it allows for the consideration of all possible values and scenarios. However, in practical applications, it is often replaced with a large but finite number for ease of use.

4. Can an infinite number of units be divided or multiplied?

No, an infinite number of units cannot be divided or multiplied by a finite number. This is because any finite number multiplied by infinity would still result in infinity, and dividing infinity by any finite number would also result in infinity.

5. Are there different sizes of infinite numbers?

Yes, there are different sizes of infinite numbers. For example, the set of all whole numbers (1, 2, 3, 4...) is considered to be infinitely larger than the set of all even numbers (2, 4, 6, 8...). This is because the second set is a subset of the first, meaning that it contains fewer elements.

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