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← Abstract Algebra
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Practice deliberately

Work through Abstract Algebra unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 7 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

Abstract Algebra coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the Abstract Algebra syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use discrete math practice questions with answers.

01

Groups

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a groups problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Subgroups

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a subgroups problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Homomorphisms

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a homomorphisms problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Quotient groups

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a quotient groups problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Rings and ideals

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a rings and ideals problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Polynomial rings

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a polynomial rings problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Field extensions

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a field extensions problem from its wording, carry out the governing method, and check that the result is reasonable.

Abstract Algebra questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 7 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: In the group of integers modulo 15 under addition, what is the order of the element 6?

Abstract Algebra · Groups

  1. 5
  2. 6
  3. 15
  4. 3
Show the worked answer
  1. The order is the smallest positive k with 6k equal to 0 modulo 15.
  2. In this group the order of an element a is 15 divided by gcd(a, 15).
  3. gcd(6, 15) = 3.
  4. 15 / 3 = 5. Check: 6 x 5 = 30, and 30 is 0 modulo 15.

Answer: 5

Question 2: A group has order 12. Which of these can NOT be the order of one of its subgroups?

Abstract Algebra · Subgroups

  1. 8
  2. 6
  3. 4
  4. 3
Show the worked answer
  1. Lagrange's theorem says a subgroup's order must divide the order of the group.
  2. The divisors of 12 are 1, 2, 3, 4, 6, and 12.
  3. 6, 4, and 3 are all on that list.
  4. 8 does not divide 12, so no subgroup can have order 8.

Answer: 8

Question 3: Define a map from the integers modulo 12 to the integers modulo 4 by reducing each class modulo 4. What is the order of the kernel?

Abstract Algebra · Homomorphisms

  1. 3
  2. 4
  3. 12
  4. 1
Show the worked answer
  1. The kernel holds every element that maps to 0.
  2. Those are the classes of 0, 4, and 8.
  3. That is 3 elements.
  4. This matches the first isomorphism theorem: 12 / 4 = 3.

Answer: 3

Question 4: What is the order of the quotient group formed by the integers modulo 12 divided by the subgroup generated by 4?

Abstract Algebra · Quotient groups

  1. 4
  2. 3
  3. 8
  4. 12
Show the worked answer
  1. The subgroup generated by 4 is {0, 4, 8}, which has 3 elements.
  2. The order of a quotient group is the group order divided by the subgroup order.
  3. 12 / 3 = 4.
  4. So the quotient group has 4 elements.

Answer: 4

Question 5: How many units does the ring of integers modulo 12 have?

Abstract Algebra · Rings and ideals

  1. 4
  2. 6
  3. 11
  4. 12
Show the worked answer
  1. An element is a unit exactly when it shares no factor with 12 other than 1.
  2. Test each: 1, 5, 7, and 11 have gcd 1 with 12.
  3. Everything else shares a factor of 2 or 3.
  4. So there are 4 units, which agrees with Euler's totient of 12.

Answer: 4

Question 6: Is x^2 + 1 irreducible over the integers modulo 3, and how many elements does the resulting quotient ring have?

Abstract Algebra · Polynomial rings

  1. Irreducible, and the quotient is a field with 9 elements
  2. Reducible, and the quotient has 9 elements
  3. Irreducible, and the quotient is a field with 3 elements
  4. Reducible, and the quotient has 6 elements
Show the worked answer
  1. A degree-2 polynomial is reducible over a field exactly when it has a root there.
  2. Test all three elements: 0 gives 1, 1 gives 2, and 2 gives 4 which is 1 modulo 3.
  3. No root, so x^2 + 1 is irreducible and the quotient ring is a field.
  4. Its elements are a + bx with a and b from three choices each, giving 3 x 3 = 9 elements.

Answer: Irreducible, and the quotient is a field with 9 elements

Question 7: What is the degree of the field extension of the rationals by both the square root of 2 and the square root of 3?

Abstract Algebra · Field extensions

  1. 4
  2. 2
  3. 6
  4. 8
Show the worked answer
  1. Adjoining the square root of 2 gives a degree 2 extension.
  2. The square root of 3 is not in that field, so adjoining it gives another degree 2 step.
  3. Degrees multiply in a tower of extensions.
  4. 2 x 2 = 4, with basis 1, root 2, root 3, and root 6.

Answer: 4

Question 8: How many elements of order 2 are in the direct product of the integers modulo 2 and the integers modulo 4?

Abstract Algebra · Groups

  1. 3
  2. 1
  3. 4
  4. 2
Show the worked answer
  1. An element (a, b) has order 2 when doubling it gives (0, 0) but the element itself is not (0, 0).
  2. In the first factor, a can be 0 or 1, and both double to 0.
  3. In the second factor, b must satisfy 2b = 0 modulo 4, so b is 0 or 2.
  4. That gives (0,0), (0,2), (1,0), (1,2). Removing the identity leaves 3 elements of order 2.

Answer: 3

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed Abstract Algebra questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1GroupsIn the group of integers modulo 15 under addition, what is the order of the element 6?5
2SubgroupsA group has order 12. Which of these can NOT be the order of one of its subgroups?8
3HomomorphismsDefine a map from the integers modulo 12 to the integers modulo 4 by reducing each class modulo 4. What is the order of the kernel?3
4Quotient groupsWhat is the order of the quotient group formed by the integers modulo 12 divided by the subgroup generated by 4?4
5Rings and idealsHow many units does the ring of integers modulo 12 have?4
6Polynomial ringsIs x^2 + 1 irreducible over the integers modulo 3, and how many elements does the resulting quotient ring have?Irreducible, and the quotient is a field with 9 elements
7Field extensionsWhat is the degree of the field extension of the rationals by both the square root of 2 and the square root of 3?4
8GroupsHow many elements of order 2 are in the direct product of the integers modulo 2 and the integers modulo 4?3

How the generated Abstract Algebra sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this Abstract Algebra practice page

What does this page cover?

It covers all 7 Abstract Algebra units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.