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Advanced Algebraic Theory

Algebra

A collection of high-level algebraic questions covering group theory, Galois theory, and field properties.

mathematics abstract-algebra theory

12 Questions Hard Ages 18+ Jul 18, 2026

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About this Study Set

This study set covers Algebra through 12 practice questions. A collection of high-level algebraic questions covering group theory, Galois theory, and field properties. Every question includes the correct answer so you can learn as you go — pick any format above to get started.

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1 Which theorem states that every non-constant polynomial with coefficients in the field of complex numbers has at least one root in the field of complex numbers?

A Fundamental Theorem of Algebra
B Lagrange's Theorem
C Sylow's Theorem
D Cauchy's Theorem

2 In group theory, what is the order of the alternating group A_n (the group of even permutations of n elements) for n ≥ 2?

A n
B n!
C n!/2
D (n-1)!

3 Which field is defined as the smallest field extension of the rational numbers that contains all the roots of a given polynomial?

A Cyclotomic field
B Splitting field
C Galois field
D Algebraic closure

4 What is the characteristic of a finite field with p^n elements, where p is a prime number?

A n
B p
C p^n
D p+n

5 According to the Abel-Ruffini theorem, why is there no general algebraic solution for polynomial equations of degree five or higher?

A Lack of sufficient field extensions
B The Galois group is not solvable
C The discriminant is always zero
D The coefficients are transcendental

6 What is the structure of the group of units of the ring of integers modulo n, denoted as (Z/nZ)ˣ, when n is a prime p?

A Cyclic group of order p
B Cyclic group of order p-1
C Symmetric group S_p
D Abelian group of order p

7 In the context of ring theory, what is a ring in which every ideal is a principal ideal called?

A Euclidean domain
B Principal Ideal Domain
C Unique Factorization Domain
D Field

8 Which condition must a ring satisfy to be classified as a division ring?

A It must be commutative
B It must contain zero divisors
C Every non-zero element must have a multiplicative inverse
D It must be a finite field

9 What is the dimension of the vector space of 3x3 matrices over a field F?

10 In Galois theory, the Galois group of a field extension is defined as the group of all automorphisms of the extension field that fix which subfield?

A The base field
B The algebraic closure
C The field of complex numbers
D The field of integers

11 Which of the following is a necessary condition for a group to be considered a simple group?

A It must be abelian
B It has no non-trivial normal subgroups
C It must be of prime order
D It must be a cyclic group

12 What is the minimum polynomial of the complex number i (the square root of -1) over the field of rational numbers?

A x - i
B x + 1
C x^2 - 1
D x^2 + 1

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