Learn Abstract Algebra unit by unit

Study algebraic structures through groups, rings, fields, and homomorphisms.

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Units at a glance

Every unit in this course, what you will be able to do once you finish it, and roughly when it lands in a 14-week schedule.

Abstract Algebra course map: 7 units
#	Unit	What you can do after it	Weeks
01	Groups	You can check whether a set with an operation forms a group and read a group table.	Weeks 1-2
02	Subgroups	You can decide whether a smaller set inside a group is itself a group, and use Lagrange's theorem on the sizes.	Weeks 3-4
03	Homomorphisms	You can map one group into another so the operation survives, then find the kernel and the image.	Weeks 5-6
04	Quotient groups	You can build a new group whose elements are whole cosets, once the subgroup is normal.	Weeks 7-8
05	Rings and ideals	You can work with a set that carries both addition and multiplication and find its ideals.	Weeks 9-10
06	Polynomial rings	You can divide, factor and test irreducibility for polynomials whose coefficients live in a ring or field.	Weeks 11-12
07	Field extensions	You can build a larger field by adding a root and measure how much larger it is with a degree.	Weeks 13-14

Abstract Algebra course units

Work in order for a complete course, or jump directly to the unit you need for an upcoming assessment.

Groups

Weeks 1-2

You can check whether a set with an operation forms a group and read a group table. You can also find the order of an element inside it.

Test closure, identity, inverses and associativity on a small set.
Find the order of the element 2 in the integers mod 6 under addition.
Explain why the integers under subtraction fail to be a group.

Worked example

Is {1, 2, 3, 4} a group under multiplication modulo 5?

The identity is 1, and 2 x 3 = 6 = 1, so 2 and 3 invert each other
4 x 4 = 16 = 1, so 4 is its own inverse

AnswerYes, it is a group of order 4

Most common mistakeIncluding 0 in the set: 0 has no multiplicative inverse mod 5, so {0, 1, 2, 3, 4} fails the group test on that one element.

Explain groups Practice groups

Subgroups

Weeks 3-4

You can decide whether a smaller set inside a group is itself a group, and use Lagrange's theorem on the sizes. You can also list the cosets it produces.

Apply the one-step subgroup test.
Use Lagrange's theorem: a subgroup's order divides the group's order.
List the left cosets of a subgroup of order 3 inside a group of order 12.

Worked example

Can a group of order 12 have a subgroup of order 5?

Lagrange: the subgroup's order has to divide 12
The divisors of 12 are 1, 2, 3, 4, 6 and 12

AnswerNo, because 5 is not on that list

Most common mistakeReading Lagrange's theorem backwards: assuming every divisor gives a subgroup, though the group A4 of order 12 has no subgroup of order 6.

Explain subgroups Practice subgroups

Homomorphisms

Weeks 5-6

You can map one group into another so the operation survives, then find the kernel and the image. You can also decide whether the map is an isomorphism.

Verify that a proposed map sends products to products.
Find the kernel as everything that lands on the identity.
Show that a trivial kernel makes the map one-to-one.

Worked example

Map each integer to its remainder mod 4. Find the kernel.

The kernel is every n whose remainder is 0
That is 0, 4, -4, 8, -8 and so on

AnswerThe kernel is the set of all multiples of 4

Most common mistakeCalling the kernel {0}: only the single integer 0 gets listed, though every other multiple of 4 also lands on the identity.

Explain homomorphisms Practice homomorphisms

Quotient groups

Weeks 7-8

You can build a new group whose elements are whole cosets, once the subgroup is normal. You can also apply the first isomorphism theorem.

Check normality by conjugating the subgroup and getting it back.
List the elements of the integers mod 4 as four cosets.
Use the first isomorphism theorem to identify a quotient with an image.

Worked example

Describe the integers mod 4 as a quotient group

Take the group of integers and the subgroup of multiples of 4
The cosets are 0 + N, 1 + N, 2 + N and 3 + N

AnswerA group of order 4

Most common mistakeBuilding a quotient by a subgroup that is not normal: in S3 the two-element subgroup generated by a single swap is not normal, so its cosets do not multiply consistently.

Explain quotient groups Practice quotient groups

Rings and ideals

Weeks 9-10

You can work with a set that carries both addition and multiplication and find its ideals. You can also spot zero divisors hiding inside it.

Check whether a subset absorbs multiplication and so forms an ideal.
Find the zero divisors in the integers mod 6.
Tell an integral domain apart from a general ring.

Worked example

Find zero divisors in the integers modulo 6

2 x 3 = 6, which is 0 mod 6
Neither 2 nor 3 is itself zero

Answer2 and 3 are zero divisors, so this ring is not an integral domain

Most common mistakeAssuming a product of zero forces one factor to be zero: that holds in the ordinary integers but fails mod 6, where 2 x 3 = 0.

Explain rings and ideals Practice rings and ideals

Polynomial rings

Weeks 11-12

You can divide, factor and test irreducibility for polynomials whose coefficients live in a ring or field. You can also apply Eisenstein's criterion.

Divide one polynomial by another and read off the remainder.
Apply Eisenstein's criterion with a chosen prime.
Explain why a polynomial can be irreducible over one field and not another.

Worked example

Is x squared + 1 irreducible over the reals? Over the complex numbers?

Over the reals it has no root, since x squared is never negative
Over the complex numbers it factors as (x + i)(x - i)

AnswerIrreducible over the reals, reducible over the complex numbers

Most common mistakeCalling a polynomial irreducible without naming the field: x squared - 2 is irreducible over the rationals but splits over the reals.

Explain polynomial rings Practice polynomial rings

Field extensions

Weeks 13-14

You can build a larger field by adding a root and measure how much larger it is with a degree. You can also multiply degrees through a chain with the tower rule.

Find the minimal polynomial of the square root of 2 over the rationals.
Compute the degree of that extension.
Use the tower rule to multiply degrees along a chain of fields.

Worked example

Find the degree of the rationals extended by the square root of 2

The square root of 2 is a root of x squared - 2, which does not factor over the rationals
A basis for the extension is 1 and the square root of 2

AnswerDegree 2

Most common mistakeReading the degree off the root symbol instead of the minimal polynomial: calling the cube root of 2 a degree-2 extension when x cubed - 2 makes it degree 3.

Explain field extensions Practice field extensions

Prepare for Abstract Algebra practice

Start with the earliest uncertain prerequisite

Before timing yourself, check whether you can explain Groups from a blank page. Then connect it to Subgroups. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.

Use tools to verify, not to choose the method for you

The Linear algebra calculator can test calculations and representations used in Abstract Algebra. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.

Know when to move to the full test

Move from Abstract Algebra practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.

Before and after the syllabus

Learn the ideas, then practise them

The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.

GuidesLinear Algebra explainedConcepts, worked methods and checks behind this course.

PracticeMixed Discrete Math questionsAnswer review after every attempt.

ReferenceFormulas for this courseConditions and a worked example for each one.

Assess

Take the complete Abstract Algebra test

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Reference

Review essential formulas

Open library →

Calculate

Use the Linear algebra calculator

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Plan

Prepare around your exam date

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Questions about the Abstract Algebra course

Where should I start?

Start with Groups if you are following the full sequence. If that unit feels automatic, open the Abstract Algebra practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.

How do I know I am ready for the course test?

You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete Abstract Algebra test only after you can correct practice errors from a blank page.

Which calculator supports this course?

The Linear algebra calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.

Learn Abstract Algebra unit by unit

Units at a glance

Abstract Algebra course units

Groups

Worked example

Subgroups

Worked example

Homomorphisms

Worked example

Quotient groups

Worked example

Rings and ideals

Worked example

Polynomial rings

Worked example

Field extensions

Worked example

Prepare for Abstract Algebra practice

Start with the earliest uncertain prerequisite

Use tools to verify, not to choose the method for you

Know when to move to the full test

Learn the ideas, then practise them

Other university courses

College Algebra

Calculus I

Calculus II

Take the complete Abstract Algebra test

Review essential formulas

Use the Linear algebra calculator

Prepare around your exam date

Questions about the Abstract Algebra course