Maths WorkshopMaths problems with full solutions

Rings and fields

33 problems · Rings, ideals, quotient rings, integral domains and field extensions.

Ex.276Checking the ring axioms
#ring axioms #examples #counterexamples
Rings and fields
High school
High school
Ex.569Fields
#field #examples #invertible elements
Rings and fields
L1
L1
Ex.570Integral domains
#integral domain #examples #cancellation
Rings and fields
L1
L1
Ex.571Zero divisors
#zero divisors #examples #rings
Rings and fields
High school
High school
Ex.1168Characteristic of a ring
#characteristic #ring #examples
Rings and fields
L1
L1
Ex.1169Euclidean division in $\mathbb{R}[X]$
#Euclidean division #polynomials #quotient and remainder
Rings and fields
L1
L1
Ex.1170Ideals
#ideal #definition #examples
Rings and fields
L1
L1
Ex.1171Ideals in $\mathbb{Z}$
#ideals #Z #principal ideal
Rings and fields
L1
L1
Ex.1172Matrix rings and zero divisors
#matrix ring #zero divisors #non-commutative
Rings and fields
L1
L1
Ex.1173The polynomial ring $R[X]$
#polynomial ring #properties #integral domain
Rings and fields
L1
L1
Ex.1174Prime ideals in $\mathbb{Z}$
#prime ideals #Z #primes
Rings and fields
L1
L1
Ex.1865Algebraic numbers
#algebraic numbers #minimal polynomial #degree
Rings and fields
M1
M1
Ex.1866Boolean rings
#Boolean ring #idempotent #commutative
Rings and fields
Olympiad
Olympiad
Ex.1867Chinese remainder theorem for rings
#Chinese remainder #coprime ideals #isomorphism
Rings and fields
M1
M1
Ex.1868Eisenstein's criterion
#Eisenstein criterion #irreducibility #polynomials
Rings and fields
M1
M1
Ex.1869$\mathbb{Z}[i]$ is Euclidean
#Euclidean domain #Gaussian integers #division
Rings and fields
M1
M1
Ex.1870Field extensions
#field extension #degree #minimal polynomial
Rings and fields
M1
M1
Ex.1871Finite fields
#finite field #order #prime power
Rings and fields
M1
M1
Ex.1872The Gaussian integers $\mathbb{Z}[i]$
#Gaussian integers #ring #norm
Rings and fields
L1
L1
Ex.1873GCD in $\mathbb{R}[X]$
#GCD #Euclidean algorithm #Bézout
Rings and fields
L1
L1
Ex.1874Irreducibility in $\mathbb{R}[X]$ and $\mathbb{C}[X]$
#irreducibility #polynomial #fields
Rings and fields
L1
L1
Ex.1875Maximal ideals in $\mathbb{R}[X]$
#maximal ideals #polynomials #irreducibility
Rings and fields
M1
M1
Ex.1876Nilpotent elements
#nilpotent #ring #quotient
Rings and fields
M1
M1
Ex.1877Principal ideal domains
#PID #principal ideal #examples
Rings and fields
M1
M1
Ex.1878The field $\mathbb{R}[X]/(X^2+1)$
#quotient ring #polynomial #complex numbers
Rings and fields
L1
L1
Ex.1879Prime characteristic
#characteristic #prime #field
Rings and fields
M1
M1
Ex.1880Prime and maximal ideals
#prime ideal #maximal ideal #quotient
Rings and fields
L1
L1
Ex.1881Quotient rings
#quotient ring #ideal #operations
Rings and fields
L1
L1
Ex.2216Irreducibility of the cyclotomic polynomial
#cyclotomic #irreducibility #field
Rings and fields
M1
M1
Ex.2217Sums of two squares via $\mathbb{Z}[i]$
#sums of squares #Gaussian integers #factorization
Rings and fields
M1
M1
Ex.2218Localization
#localization #fraction field #ring
Rings and fields
M1
M1
Ex.2219The spectrum of a ring (statement)
#spectrum #prime ideals #Zariski topology
Rings and fields
M1
M1
Ex.2220Unique factorization domains
#UFD #unique factorization #examples
Rings and fields
M1
M1