Mathematics Qualifying Exam University of British Columbia September 2, 2010
... 3. Let A be a square matrix with all diagonal entries equal to 2, all entries directly above or below the main diagonal equal to 1, and all other entries equal to 0. Show that every eigenvalue of A is a real number strictly between 0 and 4. ...
... 3. Let A be a square matrix with all diagonal entries equal to 2, all entries directly above or below the main diagonal equal to 1, and all other entries equal to 0. Show that every eigenvalue of A is a real number strictly between 0 and 4. ...
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... A polynomial that cannot be factored into two lower-degree polynomials with real number coefficients is irreducible over the reals. Theorem 1.5: Any constant or linear polynomial is irreducible over the reals. A quadratic polynomial is irreducible over the reals iff its discriminant is negative. No ...
... A polynomial that cannot be factored into two lower-degree polynomials with real number coefficients is irreducible over the reals. Theorem 1.5: Any constant or linear polynomial is irreducible over the reals. A quadratic polynomial is irreducible over the reals iff its discriminant is negative. No ...
1 Principal Ideal Domains
... We’ve already proved all of these - the only thing we needed the Euclidean Domain property for was a way to find that GCD. In PIDs, we at least are guaranteed that the GCD will exist, although it may in general be hard to find. So, what’s so great about PIDs? Well, number one is the following. Theor ...
... We’ve already proved all of these - the only thing we needed the Euclidean Domain property for was a way to find that GCD. In PIDs, we at least are guaranteed that the GCD will exist, although it may in general be hard to find. So, what’s so great about PIDs? Well, number one is the following. Theor ...
Fundamental Notions in Algebra – Exercise No. 10
... (b) R has a faithful semi-simple module; (c) R is a subdirect product of primitive rings (d) J(R) = 0. 2. (a) Show that a ring R is primitive if and only if it contains a left ideal I which does not contain a non-zero two-sided ideal. (b) Show that a commutative ring R is primitive if and only if it ...
... (b) R has a faithful semi-simple module; (c) R is a subdirect product of primitive rings (d) J(R) = 0. 2. (a) Show that a ring R is primitive if and only if it contains a left ideal I which does not contain a non-zero two-sided ideal. (b) Show that a commutative ring R is primitive if and only if it ...
Irish Intervarsity Mathematics Competition 2002 University College Dublin Time allowed: Three hours
... 5. If f (n) = an2 + bn + c, where a, b, c and n are all positive integers, show that there exists a value of n for which f (n) is not a prime number. 6. What is the area of a smallest rectangle into which squares of areas 12 , 22 , 32 , 42 , 52 , 62 , 72 , 82 textand92 can simultaneously be fitted w ...
... 5. If f (n) = an2 + bn + c, where a, b, c and n are all positive integers, show that there exists a value of n for which f (n) is not a prime number. 6. What is the area of a smallest rectangle into which squares of areas 12 , 22 , 32 , 42 , 52 , 62 , 72 , 82 textand92 can simultaneously be fitted w ...
The Fundamental Theorem of Algebra from a Constructive Point of
... Recap: Given a monic, irreducible polynomial g(y) with integer coefficients, the field obtained by adjoining one root of g to the field Q of rational numbers is by definition the field Q[y] mod g(y). It may well contain only one root of g, though, and we want deg g roots. Let me pause a moment to re ...
... Recap: Given a monic, irreducible polynomial g(y) with integer coefficients, the field obtained by adjoining one root of g to the field Q of rational numbers is by definition the field Q[y] mod g(y). It may well contain only one root of g, though, and we want deg g roots. Let me pause a moment to re ...
Chapter 1 (as PDF)
... • A ring is called a division ring (or skew field) if the non-zero elements form a group under ∗. • A commutative division ring is called a field. Example 2.3 • the integers (Z, +, ∗) form an integral domain but not a field; • the rationals (Q, +, ∗), reals (R, +, ∗) and complex numbers (C, +, ∗) fo ...
... • A ring is called a division ring (or skew field) if the non-zero elements form a group under ∗. • A commutative division ring is called a field. Example 2.3 • the integers (Z, +, ∗) form an integral domain but not a field; • the rationals (Q, +, ∗), reals (R, +, ∗) and complex numbers (C, +, ∗) fo ...
Ring class groups and ring class fields
... group to be IK (f)/PK,Z (f), which is naturally isomorphic to the group of ideals of O prime to f modulo principal ideals of O prime to f. We define the class number hO of O to be the order of the ring class group. The proposition gives a natural isomorphism between the ideal class group of O and th ...
... group to be IK (f)/PK,Z (f), which is naturally isomorphic to the group of ideals of O prime to f modulo principal ideals of O prime to f. We define the class number hO of O to be the order of the ring class group. The proposition gives a natural isomorphism between the ideal class group of O and th ...
Algebra I - Mr. Garrett's Learning Center
... • Variable – A variable is a letter or symbol that represents a number (unknown quantity). • 8 + n = 12 ...
... • Variable – A variable is a letter or symbol that represents a number (unknown quantity). • 8 + n = 12 ...
Here`s a handout - Bryn Mawr College
... 1. Show that if x, y, and z are elements of any field, then these things are true: a. If x + y = x + z, then y = z. b. (–1) ∙ x = (–x) (Show that (–1) ∙ x has the property that –x is supposed to have; but –x is supposed to be the unique number with that property) c. (–x) ∙ (–y) = xy d. If xy = 0, th ...
... 1. Show that if x, y, and z are elements of any field, then these things are true: a. If x + y = x + z, then y = z. b. (–1) ∙ x = (–x) (Show that (–1) ∙ x has the property that –x is supposed to have; but –x is supposed to be the unique number with that property) c. (–x) ∙ (–y) = xy d. If xy = 0, th ...
Math 562 Spring 2012 Homework 4 Drew Armstrong
... [In general, given any field F we define its prime subfield F 0 ⊆ F as the intersection of all subfields — equivalently, F 0 is the subfield generated by 1F . It’s a general fact that the prime subfield is isomorphic to either Q or Z/(p), depending on the characteristic of F . You just proved the ch ...
... [In general, given any field F we define its prime subfield F 0 ⊆ F as the intersection of all subfields — equivalently, F 0 is the subfield generated by 1F . It’s a general fact that the prime subfield is isomorphic to either Q or Z/(p), depending on the characteristic of F . You just proved the ch ...
Lesson 2 – The Unit Circle: A Rich Example for
... identify map and the conjugation map . Note that the set of elements in fixed by is just . For general Galois extensions ...
... identify map and the conjugation map . Note that the set of elements in fixed by is just . For general Galois extensions ...
4. Lecture 4 Visualizing rings We describe several ways - b
... norm, every non-zero element of R/(a) √ is represented by a unit, so in particular R/(a) is a field. In particular Z[(1 + −19/2)] is not Euclidean, even though it is an imaginary quadratic number field with unique factorization.) Moreover not many rings can be embedded inside Cn (though this does ap ...
... norm, every non-zero element of R/(a) √ is represented by a unit, so in particular R/(a) is a field. In particular Z[(1 + −19/2)] is not Euclidean, even though it is an imaginary quadratic number field with unique factorization.) Moreover not many rings can be embedded inside Cn (though this does ap ...
