Rationals Mod Ideal & Prime: Isomorphic to Z_p

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In summary, the conversation discusses how to show that the ring of rational numbers with reduced form denominators not divisible by a prime p, modulo an ideal consisting of elements with numerators divisible by p, is isomorphic to Z_p. The conversation includes attempts at constructing an isomorphism and ultimately concludes by proving that the ring is in fact a field with p elements.
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faradayslaw
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Homework Statement


Show that the ring of rational numbers whose reduced form denominator is not divisble by a prime, p, mod an ideal the set of elements of the above set whose numerators are divisible by p is isomorphic to Z_p

Homework Equations


The Attempt at a Solution


It seems very trivial: Use 1st homomorphism theorem with phi(a/b) = a(modp), but I am having a hard time showing that such a mapping is actually a homomorphism additively. I.E., phi(a/b + c/d) = phi(ad+bc/bd) = ad+bc mod(p) =/= a modp + b modp = phi(a/b) + phi (c/d).

I am stuck here and any help would be appreciated.

Thanks,
 
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  • #2


Instead of constructing an isomorphism directly, maybe you can show that it's a field with p elements. This would imply it as well.
 
  • #3


Worked well, I showed the following:
I is maximal in R, since if we have N an ideal with N=/= I then, there is a/b in N with p|\a -> a=/= 0, for if a=0, p|a. Then, there exists a^-1 in Q s.t. a*a^-1 = a^-1 * a = 1. b=/= 0 -> there exists b^-1 with the same property. Since N is an ideal, (a^-1/b^-1)*(a/b) is in N -> 1 is in N -> N=R -> I is maximal -> R/I is a field (R is commutative with 1).

Then, the number of distinct additive cosets r+I is precisely p QED

Thanks
 

1. What is the definition of a rational number?

A rational number is a number that can be expressed as the ratio of two integers, where the denominator is not equal to zero. In other words, it can be written in the form a/b, where a and b are integers.

2. What is an ideal in abstract algebra?

In abstract algebra, an ideal is a subset of a ring that satisfies certain properties, such as closure under addition and multiplication by elements of the ring. In the context of rationals mod ideal and prime, the ideal refers to a set of rational numbers that are equivalent to 0 when reduced modulo a prime number.

3. What does it mean for two structures to be isomorphic?

Two structures are said to be isomorphic if there exists a mapping between them that preserves the structure and properties of the elements. In the case of rationals mod ideal and prime, this means that the two structures have the same underlying operations and behave in the same way.

4. What is the significance of Z_p in this context?

Z_p refers to the set of integers modulo a prime number p. In the context of rationals mod ideal and prime, it represents the set of rational numbers that are equivalent to 0 when reduced modulo p. This is important because it allows us to define a finite group structure on the set of rationals mod ideal and prime.

5. How is this concept relevant in mathematics?

The concept of rationals mod ideal and prime being isomorphic to Z_p is relevant in various areas of mathematics, such as abstract algebra and number theory. It helps us understand the properties and structure of rational numbers, and also has applications in cryptography and coding theory.

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