Stories with Tag Field Theory

• From Finite Integral Domains to Finite Fields

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  Posted: 2025-05-26 13:43:45

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  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.

  • finite fields
  • integral domains
  • finite integral domains
  • Abstract Algebra
  • ring theory
  • Field Theory
  • mathematics
  • algebra
  • number theory
  • commutative algebra

  Summary of Comments ( 13 )
  https://news.ycombinator.com/item?id=44097362

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  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.

• Lemma for the Fundamental Theorem of Galois Theory

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  Posted: 2025-03-15 15:38:22

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  This post presents a simplified, self-contained proof of a key lemma used in proving the Fundamental Theorem of Galois Theory. This lemma establishes a bijection between intermediate fields of a Galois extension and subgroups of its Galois group. The core idea involves demonstrating that for a finite Galois extension K/F and an intermediate field E, the fixed field of the automorphism group fixing E (denoted as Inv(Gal(K/E)) is E itself. The proof leverages the linear independence of field automorphisms and constructs a polynomial whose roots distinguish elements within and outside of E, thereby connecting the field structure to the group structure. This direct approach avoids more complex machinery sometimes used in other proofs, making the fundamental theorem's core connection more accessible.

  This blog post meticulously dissects a crucial lemma that underpins the Fundamental Theorem of Galois Theory, a cornerstone of abstract algebra. The theorem establishes a profound connection between intermediate fields of a Galois extension and subgroups of its Galois group. This specific lemma, while seemingly technical, plays a vital role in demonstrating the injectivity of the correspondence described in the Fundamental Theorem. In essence, it proves that distinct subgroups of the Galois group must fix distinct intermediate fields.

  The author begins by carefully setting the stage, reminding the reader of the context within Galois theory. We are considering a field extension K of F, specifically a Galois extension, meaning it's a normal and separable extension. The Galois group Gal(K/F) consists of all automorphisms of K that leave the base field F fixed. The lemma focuses on the relationship between subgroups H of this Galois group and the intermediate fields L, where F ⊆ L ⊆ K.

  The statement of the lemma itself asserts that if we have two distinct subgroups H₁ and H₂ of Gal(K/F), then the fixed fields of these subgroups, denoted as K^( H₁) and K^( H₂), respectively, must also be distinct. The fixed field of a subgroup H is the set of all elements in K that remain unchanged under every automorphism in H. Thus, the lemma claims that different subgroups fix different parts of the extension field K.

  The proof proceeds by contradiction. We assume that the fixed fields are equal, K^( H₁) = K^( H₂), and aim to show that this implies H₁ = H₂, contradicting our initial assumption of distinct subgroups. The author leverages the fact that any automorphism in H₁ will necessarily fix K^( H₁) and, by our assumption, also fix K^( H₂). This allows us to conclude that any element of H₁ belongs to the subgroup that fixes K^( H₂). Due to the Galois extension context, we know that H₂ is precisely the subgroup that fixes K^( H₂), implying that H₁ is a subset of H₂.

  The argument is then symmetrically applied: because K^( H₁) = K^( H₂), any automorphism in H₂ fixes K^( H₂), and therefore also fixes K^( H₁). Consequently, H₂ is a subset of H₁, which is the subgroup that fixes K^( H₁). Since H₁ is a subset of H₂ and H₂ is a subset of H₁, we deduce that H₁ and H₂ must be the same subgroup, a contradiction.

  Therefore, the initial assumption that distinct subgroups have the same fixed field is false, establishing the lemma. This result is a key ingredient in the proof of the Fundamental Theorem of Galois Theory, ensuring that the mapping between subgroups and intermediate fields is indeed injective. The post concludes by highlighting the lemma's importance within the broader framework of Galois theory.

  • Galois Theory
  • Fundamental Theorem of Galois Theory
  • mathematics
  • Abstract Algebra
  • Field Theory
  • Group Theory
  • Lemma
  • Proof
  • Theorem
  • Algebraic Extensions
  • Normal Extensions
  • Separable Extensions
  • Automorphisms
  • Fixed Fields

  Summary of Comments ( 0 )
  https://news.ycombinator.com/item?id=43373196

  Verbose tl;dr

  Hacker News users discuss the linked blog post explaining a lemma used in the proof of the Fundamental Theorem of Galois Theory. Several commenters appreciate the clear explanation of a complex topic, with one pointing out how helpful the visualization and step-by-step breakdown of the proof is. Another commenter highlights the author's effective use of simple examples to illustrate the core concepts. Some discussion revolves around different approaches to teaching and understanding Galois theory, including alternative proofs and the role of intuition versus rigor. One user mentions the value of seeing multiple perspectives on the same concept to solidify understanding. The overall sentiment is positive, praising the author's pedagogical approach to demystifying a challenging area of mathematics.

  The Hacker News post titled "Lemma for the Fundamental Theorem of Galois Theory" sparked a brief discussion with a few insightful comments. No one overtly disagreed with the premise of the linked article, but rather expanded on its context and implications.

  One commenter pointed out the significance of the lemma by highlighting that it establishes a connection between the intermediate fields of a Galois extension and the subgroups of its Galois group. This connection, they explain, is crucial for the Fundamental Theorem of Galois Theory, which establishes a deeper correspondence between these intermediate fields and subgroups, going beyond just a bijection and relating their structure and properties as well. This comment effectively underscores the lemma's role as a foundational building block for the broader theorem.

  Another commenter delves into more specific details, mentioning that the lemma facilitates a clearer understanding of the bijective nature of the Galois correspondence. They mention that some formulations of the Fundamental Theorem of Galois Theory include this lemma as an integral part, demonstrating different approaches to presenting the theorem. This comment highlights variations in how the theorem and its supporting components are presented in different mathematical texts and learning materials.

  A further comment discusses the pedagogical implications, noting how the lemma simplifies the proof of the Fundamental Theorem of Galois Theory by separating out a key argument. This streamlined approach is viewed as beneficial for understanding the overall logic and flow of the proof.

  Finally, another contributor shifts the focus to the overall context of the Fundamental Theorem of Galois Theory within abstract algebra, emphasizing its importance and depth. This comment emphasizes the broader significance of the topic within its mathematical field.

  In summary, while limited in number, the comments on the Hacker News post provide valuable perspectives on the presented lemma, including its importance within the proof of the Fundamental Theorem of Galois Theory, different approaches to its presentation, and its broader mathematical significance.