This post demonstrates that every finite integral domain is also a field. It begins by establishing that a finite integral domain possesses the cancellation property, meaning if ab = ac and a is nonzero, then b = c. Leveraging this property, the author then shows that repeated multiplication by a nonzero element a within the finite domain must eventually yield a cycle, since only finitely many elements exist. By analyzing the elements within this cycle and again using the cancellation property, the author proves the existence of a multiplicative identity and multiplicative inverses for all nonzero elements. Thus, the finite integral domain fulfills all field axioms, confirming the initial assertion.
This blog post elucidates the fascinating relationship between finite integral domains and finite fields, ultimately demonstrating that they are, in fact, one and the same. It begins by laying out the fundamental definitions necessary for the ensuing argument. An integral domain is defined as a commutative ring with unity and no zero divisors, meaning the product of any two non-zero elements is also non-zero. A field, meanwhile, is defined as a commutative ring with unity where every non-zero element has a multiplicative inverse.
The core objective of the post is to prove that every finite integral domain is also a field. The proof leverages a crucial property of finite sets: any injective function from a finite set to itself is also surjective. This principle is applied to the multiplication operation within the finite integral domain.
The post meticulously constructs the argument as follows: Let R be a finite integral domain. We choose a non-zero element 'a' from R. We then define a function f from R to itself, where f(x) is equal to the product of 'a' and 'x' (i.e., f(x) = ax). Because R is an integral domain and 'a' is non-zero, this function f is injective. This follows from the property of integral domains that lack zero divisors: If ax = ay, then a(x - y) = 0. Since 'a' is non-zero, x - y must be zero, meaning x = y. Therefore, f is injective.
Now, since R is a finite set, the injectivity of f implies its surjectivity. This means that for every element 'y' in R, there exists an element 'x' in R such that f(x) = y, or equivalently, ax = y. This holds true for all elements 'y' in R, including the multiplicative identity '1'. Consequently, there exists an element 'x' in R such that ax = 1. This 'x' is the multiplicative inverse of 'a'.
Because 'a' was an arbitrary non-zero element of R, this demonstrates that every non-zero element in R possesses a multiplicative inverse. Thus, R, a finite integral domain, satisfies the criteria to be a field. Therefore, the post concludes that any finite integral domain is inherently a finite field. This establishes the equivalence of finite integral domains and finite fields, a fundamental result in abstract algebra.
Summary of Comments ( 13 )
https://news.ycombinator.com/item?id=44097362
Hacker News users generally praised the article for its clear explanation of a complex mathematical concept. Several commenters appreciated the author's approach of starting with familiar concepts like integers and polynomials, then gradually introducing more abstract ideas. One commenter highlighted the helpful use of concrete examples throughout the explanation. Another pointed out the pedagogical value of showing the construction of finite fields, rather than just stating their existence. A few comments mentioned related concepts, like the use of finite fields in cryptography and coding theory, and the difference between integral domains and fields. Overall, the sentiment was positive, with commenters finding the article to be well-written and insightful.
The Hacker News post titled "From Finite Integral Domains to Finite Fields," linking to an article on susam.net explaining the relationship between finite integral domains and finite fields, generated a modest discussion thread.
Several commenters praised the clarity and conciseness of the explanation. One user appreciated the article's accessibility, highlighting how it presented a clear path from basic definitions to the final conclusion. Another commenter echoed this sentiment, specifically noting how the article effectively explained the concept without requiring advanced mathematical background. The gentle pace and clear build-up of the proof were pointed out as particularly helpful.
A few commenters discussed the practical applications of finite fields, mentioning their use in cryptography and coding theory. One user specifically mentioned Reed-Solomon codes as an example, connecting the theoretical concepts presented in the article to a real-world application. Another commenter pointed out the significance of finite fields in computer science, highlighting their role in checksum algorithms like CRC32.
One commenter provided a slightly more technical perspective, suggesting that viewing finite integral domains as finite-dimensional vector spaces over their prime subfield might offer another insightful approach to understanding the relationship between finite integral domains and finite fields.
The discussion also briefly touched upon the broader topic of abstract algebra. One commenter praised the way abstract algebra reveals the underlying structures of seemingly different mathematical objects, connecting the topic of the article to a larger mathematical context.
While the overall volume of comments was not extensive, the discussion was generally positive and focused on the article's pedagogical value and the importance of finite fields in various applications. No particularly dissenting or controversial opinions were expressed.