8. Prime Factorization and Primary Decompositions
... neither 1 + 5 i nor 1 − 5 i. Hence 2 is not prime. But one can show that 2 is irreducible in R,√ and thus R √ is not a unique factorization domain [G1, Example 11.4]. In fact, 2 · 3 = (1 + 5 i)(1 − 5 i) are two decompositions of the same number 6 that do not agree up to permutation and units. It fol ...
... neither 1 + 5 i nor 1 − 5 i. Hence 2 is not prime. But one can show that 2 is irreducible in R,√ and thus R √ is not a unique factorization domain [G1, Example 11.4]. In fact, 2 · 3 = (1 + 5 i)(1 − 5 i) are two decompositions of the same number 6 that do not agree up to permutation and units. It fol ...
MA3A6 Algebraic Number Theory
... Notation 1.1.1. Let K and L be fields. If K is a subfield of L, we say L is a field extension of K, and we write L | K. For instance, C | Q is a field extension, as is C | R. Definition 1.1.2. Let L | K be a field extension, and let α ∈ L. We say α is algebraic over K if there exists a nonzero polyn ...
... Notation 1.1.1. Let K and L be fields. If K is a subfield of L, we say L is a field extension of K, and we write L | K. For instance, C | Q is a field extension, as is C | R. Definition 1.1.2. Let L | K be a field extension, and let α ∈ L. We say α is algebraic over K if there exists a nonzero polyn ...
Commutative Algebra Notes Introduction to Commutative Algebra
... We will do this by induction in fact! The statement is certainly true when n is 1. We now prove it for n > 0. Assume that the result is true for n − 1. Suppose a * pi for all i = 1 . . . n. Then for each i the remaining n − 1 ideals satisfy the induction hypothesis so we can say a* ...
... We will do this by induction in fact! The statement is certainly true when n is 1. We now prove it for n > 0. Assume that the result is true for n − 1. Suppose a * pi for all i = 1 . . . n. Then for each i the remaining n − 1 ideals satisfy the induction hypothesis so we can say a* ...
Chapter 8 - U.I.U.C. Math
... finitely many elements x1 , . . . , xn in A that generate A over k in the sense that every element of A is a polynomial in the xi . Equivalently, A is a homomorphic image of the polynomial ring k[X1 , . . . , Xn ] via the map determined by Xi → xi , i = 1, . . . , n. There exists a subset {y1 , . . . ...
... finitely many elements x1 , . . . , xn in A that generate A over k in the sense that every element of A is a polynomial in the xi . Equivalently, A is a homomorphic image of the polynomial ring k[X1 , . . . , Xn ] via the map determined by Xi → xi , i = 1, . . . , n. There exists a subset {y1 , . . . ...
The discriminant
... simplest case of a quadratic equation, the discriminant tells us the behavior of solution, and of course, even its square roots gives us the solutions. To some extent the same is true for cubic equations, and the higher the degree of an equation less the influence of the discriminant is, but it alwa ...
... simplest case of a quadratic equation, the discriminant tells us the behavior of solution, and of course, even its square roots gives us the solutions. To some extent the same is true for cubic equations, and the higher the degree of an equation less the influence of the discriminant is, but it alwa ...
Ring Theory
... mechanisms by which the subject progresses. The definition of a ring consists of a list of technical properties, but the motivation for this definition is the ubiquity of objects having these properties, like the ones in Section 1.1. When making a definition like that of a ring (or group or vector s ...
... mechanisms by which the subject progresses. The definition of a ring consists of a list of technical properties, but the motivation for this definition is the ubiquity of objects having these properties, like the ones in Section 1.1. When making a definition like that of a ring (or group or vector s ...
Notes on Ring Theory
... of integers, one defined what should ideally behave like numbers, hence the word “ideal”. Given an element r in a ring R, let a, b ∈ R and note that any (two-sided) ideal containing r must also contain arb. If 1 ∈ R, then a, b are allowed to be any integers (or really their images in the ring). If 1 ...
... of integers, one defined what should ideally behave like numbers, hence the word “ideal”. Given an element r in a ring R, let a, b ∈ R and note that any (two-sided) ideal containing r must also contain arb. If 1 ∈ R, then a, b are allowed to be any integers (or really their images in the ring). If 1 ...
1 - Evan Chen
... We begin with several definitions. Definition 5.2. The center of a group G, called Z(G), is the set of elements g ∈ G which commute with all elements of G. Suppose we want to generalize to any A ⊆ G instead of all elements. Definition 5.3. The centralizer of A ⊆ G is the set of g ∈ G which commute w ...
... We begin with several definitions. Definition 5.2. The center of a group G, called Z(G), is the set of elements g ∈ G which commute with all elements of G. Suppose we want to generalize to any A ⊆ G instead of all elements. Definition 5.3. The centralizer of A ⊆ G is the set of g ∈ G which commute w ...
6.6. Unique Factorization Domains
... Proof. Suppose that f (x) ∈ R[x] is primitive in R[x] and irreducible in F [x]. If f (x) = a(x)b(x) in R[x], then one of a(x) and b(x) must be a unit in F [x], so of degree 0. Suppose without loss of generality that a(x) = a0 ∈ R. Then a0 divides all coefficients of f (x), and, because f (x) is prim ...
... Proof. Suppose that f (x) ∈ R[x] is primitive in R[x] and irreducible in F [x]. If f (x) = a(x)b(x) in R[x], then one of a(x) and b(x) must be a unit in F [x], so of degree 0. Suppose without loss of generality that a(x) = a0 ∈ R. Then a0 divides all coefficients of f (x), and, because f (x) is prim ...
IDEAL FACTORIZATION 1. Introduction We will prove here the
... Notice this proof is close to the proof that every integer > 1 is a product of primes. Why didn’t we prove in Lemma 3.1 that every nonzero proper ideal equals (rather than merely contains) a product of nonzero prime ideals? Because we do not know (yet) that every non-prime ideal in OK is a product o ...
... Notice this proof is close to the proof that every integer > 1 is a product of primes. Why didn’t we prove in Lemma 3.1 that every nonzero proper ideal equals (rather than merely contains) a product of nonzero prime ideals? Because we do not know (yet) that every non-prime ideal in OK is a product o ...
Ring Theory
... This tells us r′ s′ is also in the coset rs + I and thus multiplication does not depend on the choice of representatives. Note though that this is true only because we assumed a two-sided ideal I, otherwise we could not have concluded, since we had to deduce that both as and rb are in I. Definition ...
... This tells us r′ s′ is also in the coset rs + I and thus multiplication does not depend on the choice of representatives. Note though that this is true only because we assumed a two-sided ideal I, otherwise we could not have concluded, since we had to deduce that both as and rb are in I. Definition ...
Appendix: Existence and Uniqueness of a Complete Ordered Field∗
... product of two negative elements of F,x × (−z) = (−x) × z = 1. The properties that guarantee that F is an ordered eld also have been established in the preceding exercises, so that the proof of this theorem is complete. So, the Dedekind eld is an ordered eld, but we have left to prove that it is ...
... product of two negative elements of F,x × (−z) = (−x) × z = 1. The properties that guarantee that F is an ordered eld also have been established in the preceding exercises, so that the proof of this theorem is complete. So, the Dedekind eld is an ordered eld, but we have left to prove that it is ...
PRIME IDEALS IN NONASSOCIATIVE RINGS
... The w-radical of the zero ideal may naturally be called the u-radical of the ring R. This concept is discussed in §4 where it is indicated that several of the expected properties of a radical hold for the w-radical. Corresponding to each element v of S3, there is an appropriate concept of v-nilpoten ...
... The w-radical of the zero ideal may naturally be called the u-radical of the ring R. This concept is discussed in §4 where it is indicated that several of the expected properties of a radical hold for the w-radical. Corresponding to each element v of S3, there is an appropriate concept of v-nilpoten ...
Math 210B. Spec 1. Some classical motivation Let A be a
... Let A be a commutative ring. We have defined the Zariski topology on the set Spec(A) of primes ideals of A by declaring the closed subsets to be those of the form V (I) = {p ⊇ I}. This is reminiscent of the classical situation where we worked with the set k n = MaxSpec(k[t1 , . . . , tn ])) for an a ...
... Let A be a commutative ring. We have defined the Zariski topology on the set Spec(A) of primes ideals of A by declaring the closed subsets to be those of the form V (I) = {p ⊇ I}. This is reminiscent of the classical situation where we worked with the set k n = MaxSpec(k[t1 , . . . , tn ])) for an a ...