Understanding Galois Fields: Why K Contains Z_p

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In summary, the book claimed that K, a subfield of \mathbb{Z}, contains Z_p. However, there is no justification for this claim and it is obvious.
  • #1
ehrenfest
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[SOLVED] galois fields

Homework Statement


Let [itex]\bar{\mathbb{Z}}_p[/itex] be the algebraic closure of [itex]\mathbb{Z}_p[/itex] and let K be the subset of [itex]\bar{\mathbb{Z}}_p[/itex] consisting of all of the zeros of [itex]x^{p^n} - x[/itex] in [itex]\bar{\mathbb{Z}}_p[/itex]. My book proved that the subset K is actually a subfield of [itex]\bar{\mathbb{Z}}_p[/itex] and that it contains p^n elements. Then, out of nowhere, it said that K contains Z_p and provided absolutely no justification for that claim. Can someone fill me in on why that is so obvious?

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The Attempt at a Solution

 
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  • #2
Because it's a field, it's going to contain 1, and as a result it will also contain 1+1, 1+1+1, ...
 
  • #3
Maybe a few general words are in order here, because you seem to be having trouble with things of this sort.

If F is a field, what is the most basic thing we can say about its elements? We know it contains 0 and 1. Consequently, it's going to contain 1+1, 1+1+1, and so on. If this process stops generating new elements at some point, then we can find two distinct positive integers n and m such that 1+1+...+1 (n times) = 1+1+...+1 (m times). From this we conclude that there is a smallest positive integer k such that 1+1+...+1 (k times) = 0 (for instance, if n>m, then 1+1+...+1 (n-m times) will give us zero). This implies that a copy of Z/kZ sits inside of F. In this case we say that F has characteristic k. On the other hand, if the process keeps generating new elements, then F is going to contain a copy of the nonnegative integers, and hence all the integers (if n is in F, then -n had better be in F too). In this case we say that F has characteristic zero.

So far we have only used the group structure of F, and not the field structure. Let's amend this now. If F contains a copy of Z, then for each n in F, 1/n is also in F. This implies that F contains a copy of Q (because m/n = 1/n + ... + 1/n (m times)). This is the most basic field of characteristic zero, and any field of characteristic zero is going to contain a copy of Q. Conversely, no field having characteristic other than zero can contain a copy of Q.

For the other case, when Z/kZ sits in F, we must have that k is a prime, for otherwise the field F contains zero divisors. This is because if p is a prime divisor of k and k!=p, then in Z/kZ, k/p and p are nonzero, and (k/p)*p=k=0. Conversely, it's easy to see that Z/pZ is a field for every prime p. Thus we conclude that a field can only have prime nonzero characteristic, and this happens iff F contains a copy of Z/pZ.

That's the long-winded version. The compact way to see all this is to consider the ring homomorphism f:Z->F given by f(n)=1+...+1 (n times). By the first isomorphism theorem, Z/kerf is going to be isomorphic to a subring of F, and thus must be an integral domain (because F is a field). This implies that kerf is a prime ideal of Z, and hence is either 0 or equal to pZ for some prime p. In the first case F will contain a copy of Z and hence of Q, and in the second case F will contain a copy of Z/pZ.

Hopefully this clears things up for you a bit.
 
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  • #4
I see, thanks.
 

Related to Understanding Galois Fields: Why K Contains Z_p

1. What is a Galois Field?

A Galois Field, also known as a finite field, is a mathematical structure used in algebraic coding theory and cryptography. It is a set of elements that follow specific rules for addition, subtraction, multiplication, and division.

2. What is the significance of the letter K in the term "K Contains Z_p"?

The letter K represents a specific Galois Field, and it is used to distinguish it from other fields. In this context, K contains Z_p means that the Galois Field K contains the prime field Z_p, which is the set of integers modulo a prime number p.

3. What is the relationship between K and Z_p in Galois Fields?

In Galois Fields, K is an extension field of Z_p. This means that all elements in Z_p are also elements in K, but K also contains additional elements that are not in Z_p. The elements in K are obtained by performing operations on the elements in Z_p.

4. Why is it important for K to contain Z_p in Galois Fields?

It is important for K to contain Z_p because Z_p serves as the building blocks for K. By including Z_p, K maintains the same algebraic properties as Z_p, making it easier to perform calculations and operations. Additionally, the inclusion of Z_p allows for the creation of Galois Field extensions, which are essential in many applications.

5. How are Galois Fields used in practical applications?

Galois Fields have various applications in different fields, including coding theory, cryptography, and error-correcting codes. They are used to create efficient and secure encryption algorithms and coding schemes. They are also used in digital signal processing and in the design of communication systems.

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