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Let be in If show that must be a unit. Show that the only units of are 1 and
Note that is in if and only if The only integer solutions to the equation are
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Let be in If show that must be a unit. Show that the only units of are 1 and
Note that is in if and only if The only integer solutions to the equation are
The Gaussian integers, are a UFD. Factor each of the following elements in into a product of irreducibles.
5
2
(a) (c)
Let be an integral domain.
Prove that is an abelian group under the operation of addition.
Show that the operation of multiplication is well-defined in the field of fractions,
Verify the associative and commutative properties for multiplication in
Prove or disprove: Any subring of a field containing 1 is an integral domain.
True.
Prove or disprove: If is an integral domain, then every prime element in is also irreducible in
Let be a field of characteristic zero. Prove that contains a subfield isomorphic to
Let be a field.
Prove that the field of fractions of denoted by is isomorphic to the set all rational expressions where is not the zero polynomial.
Let and be polynomials in Show that the set of all rational expressions is isomorphic to the field of fractions of We denote the field of fractions of by
Let be prime and denote the field of fractions of by Prove that is an infinite field of characteristic
Prove that the field of fractions of the Gaussian integers, is
Let and be in Prove that
A field is called a if it has no proper subfields. If is a subfield of and is a prime field, then is a of
Prove that every field contains a unique prime subfield.
If is a field of characteristic 0, prove that the prime subfield of is isomorphic to the field of rational numbers,
If is a field of characteristic prove that the prime subfield of is isomorphic to
Let
Prove that is an integral domain.
Find all of the units in
Determine the field of fractions of
Prove that is a Euclidean domain under the Euclidean valuation
Let be a UFD. An element is a if and and is divisible by any other element dividing both and
If is a PID and and are both nonzero elements of prove there exists a unique greatest common divisor of and up to associates. That is, if and are both greatest common divisors of and then and are associates. We write for the greatest common divisor of and
Let be a PID and and be nonzero elements of Prove that there exist elements and in such that
Let be an integral domain. Define a relation on by if and are associates in Prove that is an equivalence relation on
Let be a Euclidean domain with Euclidean valuation If is a unit in show that
Let be a Euclidean domain with Euclidean valuation If and are associates in prove that
Let with a unit. Then Similarly,
Show that is not a unique factorization domain.
Show that 21 can be factored in two different ways.
Prove or disprove: Every subdomain of a UFD is also a UFD.
An ideal of a commutative ring is said to be if there exist elements in such that every element can be written as for some in Prove that satisfies the ascending chain condition if and only if every ideal of is finitely generated.
Let be an integral domain with a descending chain of ideals Suppose that there exists an such that for all A ring satisfying this condition is said to satisfy the , or . Rings satisfying the DCC are called , after Emil Artin. Show that if satisfies the descending chain condition, it must satisfy the ascending chain condition.
Let be a commutative ring with identity. We define a of to be a subset such that and if
Define a relation on by if there exists an such that Show that is an equivalence relation on
Let denote the equivalence class of and let be the set of all equivalence classes with respect to Define the operations of addition and multiplication on by
respectively. Prove that these operations are well-defined on and that is a ring with identity under these operations. The ring is called the of with respect to
Show that the map defined by is a ring homomorphism.
If has no zero divisors and show that is one-to-one.
Prove that is a prime ideal of if and only if is a multiplicative subset of
If is a prime ideal of and show that the ring of quotients has a unique maximal ideal. Any ring that has a unique maximal ideal is called a .