Unlock hundreds more features
Save your Quiz to the Dashboard
View and Export Results
Use AI to Create Quizzes and Analyse Results

OR
Sign inSign in with Facebook
Sign inSign in with Google
Quizzes > Mathematics & Statistics

Algebraic Number Theory Quiz

Free Practice Quiz & Exam Preparation

Difficulty: Moderate
Questions: 15
Study OutcomesAdditional Reading
3D voxel art representation of the Algebraic Number Theory course

Dive into our engaging Algebraic Number Theory practice quiz, designed for students eager to master key topics such as valuation theory, ideal theory, and units in algebraic number fields. This quiz also covers essential concepts in ramification, function fields, and local class field theory, providing a comprehensive review to boost your confidence before diving into more advanced studies.

An algebraic number field is defined as a finite extension of which field?
The field of rational numbers (Q)
The field of real numbers (R)
The field of complex numbers (C)
The ring of integers (Z)
An algebraic number field is defined as a finite extension of Q, the field of rational numbers. This definition is fundamental in algebraic number theory.
Which of the following properties is characteristic of a non-archimedean valuation?
v(x+y) = v(x) + v(y)
v(x+y) ≥ min{v(x), v(y)}
v(x+y) ≤ v(x) + v(y)
v(x+y) = max{v(x), v(y)}
The ultrametric inequality, v(x+y) ≥ min{v(x), v(y)}, is a defining property of non-archimedean valuations. It distinguishes them from the standard triangle inequality found in archimedean valuations.
Which of the following best describes the ring of integers in an algebraic number field?
The set of all elements that are algebraic integers (the integral closure of Z in the field)
A maximal subfield of the number field
The group of units of the field
Elements satisfying a non-monic polynomial
The ring of integers in a number field is defined as the integral closure of Z in that field, which is equivalent to the set of all algebraic integers in the field. This ring is essential for studying ideal factorization.
In ideal factorization within a number field, what characterizes a ramified prime?
It splits into distinct prime ideals
It remains inert and does not split
It appears in the factorization with an exponent greater than 1
It becomes a unit in the ring of integers
A prime is ramified in an extension when at least one prime ideal in its factorization appears with an exponent greater than one. This increased multiplicity signifies irregular behavior compared to unramified primes.
Local class field theory primarily studies which type of fields?
Global fields such as Q
Local fields, including completions like Q_p
Fields with trivial valuation
Algebraically closed fields
Local class field theory focuses on local fields, which are completions of global fields with respect to non-archimedean valuations, such as Q_p. It explores the abelian extensions and reciprocity laws within these fields.
Let v be a discrete valuation on a field K. Which statement is true regarding its valuation ring R_v?
R_v is a field
R_v is a principal ideal domain with a unique nonzero maximal ideal
R_v has infinitely many maximal ideals
R_v is not integrally closed
The valuation ring of a discrete valuation, known as a discrete valuation ring (DVR), is a principal ideal domain with a unique nonzero maximal ideal. This property is crucial for understanding local properties of number fields.
In the context of ideal theory, what does the term 'ideal class group' refer to?
The group of all ideals in the ring of integers
The quotient of the fractional ideals by the principal ideals
The subgroup of invertible ideals
The set of maximal ideals in the ring of integers
The ideal class group is defined as the group of fractional ideals modulo the subgroup of principal ideals. This concept measures the failure of unique factorization in the ring of integers of a number field.
What is the significance of the Dirichlet Unit Theorem?
It determines the splitting behavior of primes
It provides the structure of the unit group in the ring of integers
It classifies all ideals in the field
It guarantees that every number field is Euclidean
The Dirichlet Unit Theorem describes the structure of the unit group in the ring of integers of a number field by showing it is a finitely generated abelian group. This theorem is pivotal in understanding the arithmetic properties of number fields.
In a finite extension of number fields, a prime ideal p in the base field is said to be unramified if:
None of its prime ideal factors occur in the extension
The residue field extension is trivial
Each prime ideal factor appears with multiplicity one
The prime ideal generates the entire ring of integers
A prime ideal is unramified in an extension if every prime ideal above it appears with multiplicity one. This ensures that the behavior of the prime remains regular under extension.
In studying function fields over finite fields, which of the following analogues is similar to number fields?
Fields with characteristic zero
Curves over finite fields
Function fields, where places correspond to valuations analogous to primes
Fields with no discrete valuation
Function fields over finite fields serve as analogues to number fields, with places playing a role similar to prime ideals. This analogy helps in transferring concepts between the arithmetic of number fields and the geometry of curves.
What does Hensel's Lemma primarily facilitate in local fields?
Decomposing ideals in global fields
Lifting solutions of polynomial congruences to solutions in complete local rings
Classifying the ideal class group
Determining the degree of field extensions
Hensel's Lemma is a key tool in local fields that allows one to lift approximate solutions of polynomial equations (typically modulo a prime power) to exact solutions within complete local rings. This is analogous to Newton's method in analysis.
In local class field theory, the Artin reciprocity law establishes a canonical isomorphism between which two groups?
The additive group of the local field and its Galois group
The multiplicative group of the local field and its abelianized Galois group
The ideal class group and the unit group
The group of local units and the ring of integers
The Artin reciprocity law provides a profound connection by establishing an isomorphism between the multiplicative group of a local field and the abelianization of its absolute Galois group. This result is central to the understanding of abelian extensions in local class field theory.
Which of the following statements about completions of number fields with respect to a nontrivial valuation is true?
The completion is always algebraically closed
The completion maintains the same field structure and is complete with respect to the induced topology
The completion loses its valuation properties entirely
The completion cannot be locally compact
Completing a number field with respect to a nontrivial valuation results in a field that preserves much of the original arithmetic structure while being complete in the corresponding topology. This property is essential in the study of local fields and their applications.
Which of the following correctly describes the relationship between ramification index, residue degree, and the degree of an extension in a number field?
The sum of the products of the ramification indices and residue degrees of all primes over a prime equals the extension degree
The product of the sums of the ramification indices and residue degrees equals the extension degree
For each prime ideal, the ramification index times the residue degree independently equals the extension degree
The extension degree is the minimum of the ramification indices and residue degrees
In an extension of number fields, the key relation is that the sum (over all primes lying above a given prime) of the products of the individual ramification indices and residue degrees equals the degree of the extension. This relation is central in understanding how primes behave in extensions.
Which concept is most directly linked to understanding the failure of unique factorization in rings of integers?
The structure of local fields
The Dirichlet Unit Theorem
The ideal class group
Hensel's Lemma
The ideal class group is a measure of the failure of unique factorization in the ring of integers of a number field. A trivial ideal class group corresponds to unique factorization, while a nontrivial group indicates the presence of non-unique factorization.
0
{"name":"An algebraic number field is defined as a finite extension of which field?", "url":"https://www.quiz-maker.com/QPREVIEW","txt":"An algebraic number field is defined as a finite extension of which field?, Which of the following properties is characteristic of a non-archimedean valuation?, Which of the following best describes the ring of integers in an algebraic number field?","img":"https://www.quiz-maker.com/3012/images/ogquiz.png"}

Study Outcomes

  1. Analyze the properties and applications of valuation theory in field extensions.
  2. Apply ideal theory to determine the structure and factorization in rings of algebraic integers.
  3. Understand and evaluate the role of units in algebraic number fields and their impact on ramification.
  4. Utilize concepts from function fields to solve problems in field theory.
  5. Synthesize elements of local class field theory to examine local-global principles in number theory.

Algebraic Number Theory Additional Reading

Embarking on the journey of Algebraic Number Theory? Here are some top-notch resources to guide you through the fascinating world of fields, valuations, and more:

  1. MIT OpenCourseWare: Topics in Algebraic Number Theory Dive into MIT's comprehensive course featuring lecture notes, problem sets, and readings covering Dedekind domains, class groups, and local fields.
  2. Notes on the Theory of Algebraic Numbers by Steve Wright This series of lecture notes delves into the elementary theory of algebraic numbers, requiring only a foundational understanding of algebra.
  3. Algebraic Number Theory Notes by Anwar Khan A detailed 110-page PDF covering topics like Diophantine equations, polynomial degrees, and unique factorization domains.
  4. Algebraic Number Theory Lecture Notes by Sidharth Hariharan Lecture notes from Imperial College London's MATH70042 course, offering insights into the latest teachings in the field.
  5. Algebraic Number Theory by Prof. Paul Gunnells Course materials from the University of Massachusetts, covering number fields, rings of integers, and zeta functions.
Make a Quiz (FREE) Copy & Edit This Quiz Create a Quiz with AI

Mathematics & Statistics Quizzes

Powered by: Quiz Maker