Book: What is ADE? Drew Armstrong Section 1: What is a number
... for solving the quadratic equation ax2 + bx + c = 0 was known since antiquity. After Gerolamo Cardano learned the complete solution of the cubic equation (before 1545), he shared this information with his student Lodovico Ferrari. Almost immediately, the younger mathematician was able to extend Card ...
... for solving the quadratic equation ax2 + bx + c = 0 was known since antiquity. After Gerolamo Cardano learned the complete solution of the cubic equation (before 1545), he shared this information with his student Lodovico Ferrari. Almost immediately, the younger mathematician was able to extend Card ...
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. ...
Document
... An expression of the form anxn + an1xn1 + ... + a0 where ai R, i = 0, 1, ... , n, and n N is a polynomial in x with real-number coefficients (or a polynomial in x over R). For each i, ai is the coefficient of xi. If i is the largest integer greater than 0 for which ai 0, the polynomial is of ...
... An expression of the form anxn + an1xn1 + ... + a0 where ai R, i = 0, 1, ... , n, and n N is a polynomial in x with real-number coefficients (or a polynomial in x over R). For each i, ai is the coefficient of xi. If i is the largest integer greater than 0 for which ai 0, the polynomial is of ...
Finite Fields
... Theorem 1.12. For any integer r such that r|(q − 1), the number of elements of order r in the multiplicative group F∗q is equal to φ(r) and F∗q has a unique subgroup of order r. Proof. By definition, the elements of order r in F∗q correspond to the roots of X r − 1 in F∗q . Let a ∈ F∗q be an element ...
... Theorem 1.12. For any integer r such that r|(q − 1), the number of elements of order r in the multiplicative group F∗q is equal to φ(r) and F∗q has a unique subgroup of order r. Proof. By definition, the elements of order r in F∗q correspond to the roots of X r − 1 in F∗q . Let a ∈ F∗q be an element ...
Introduction to finite fields
... K0 . We thus have that: (1) K0 ⊆ Fi , (2) P (X) factors into linear factors in Fi [X], and (3) No strict subfield satisfies both (1) and (2). We can then proceed by induction. Now suppose F1 , F2 are finite fields of cardinality q = pn , where p is prime. Set K = Fp , and we have that K ⊆ Fi . Now w ...
... K0 . We thus have that: (1) K0 ⊆ Fi , (2) P (X) factors into linear factors in Fi [X], and (3) No strict subfield satisfies both (1) and (2). We can then proceed by induction. Now suppose F1 , F2 are finite fields of cardinality q = pn , where p is prime. Set K = Fp , and we have that K ⊆ Fi . Now w ...
MS Word
... a. Is it a ring? Why or why not? Yes, this is a ring. All group properties hold for addition; multiplication is associative and closed. These are the required ring properties. b. A commutative ring? Why or why not? Yes; multiplying any two elements modular 18 will yield the same result no matter wha ...
... a. Is it a ring? Why or why not? Yes, this is a ring. All group properties hold for addition; multiplication is associative and closed. These are the required ring properties. b. A commutative ring? Why or why not? Yes; multiplying any two elements modular 18 will yield the same result no matter wha ...
Exercises 5 5.1. Let A be an abelian group. Set A ∗ = HomZ(A,Q/Z
... (b) Each one of the following R-modules is isomorphic to L (A, B; C): N i. HomR (A R B, C); ii. HomR (A, HomR (B, C)); iii. HomR (B, HomR (A, C)). 5.4. An algebra A over a field K is called a division algebra, if A is a division ring. Give an example of noncommutative division algebra over R. 5.5. L ...
... (b) Each one of the following R-modules is isomorphic to L (A, B; C): N i. HomR (A R B, C); ii. HomR (A, HomR (B, C)); iii. HomR (B, HomR (A, C)). 5.4. An algebra A over a field K is called a division algebra, if A is a division ring. Give an example of noncommutative division algebra over R. 5.5. L ...
Algebraic Expressions and Terms
... Algebraic Expressions When variables are used with other numbers, parentheses, or operations, they create an algebraic expression. a + 2 (a) (b) 3m + 6n - 6 ...
... Algebraic Expressions When variables are used with other numbers, parentheses, or operations, they create an algebraic expression. a + 2 (a) (b) 3m + 6n - 6 ...
On sum-sets and product-sets of complex numbers
... A similar argument works for quaternions and for other hypercomplex numbers. In general, if T and Q are sets of similarity transformations and A is a set of points in space such that from any quadruple (t(p1 ), t(p2 ), q(p1 ), q(p2 )) the elements t ∈ T , q ∈ Q, and p1 6= p2 ∈ A are uniquely determi ...
... A similar argument works for quaternions and for other hypercomplex numbers. In general, if T and Q are sets of similarity transformations and A is a set of points in space such that from any quadruple (t(p1 ), t(p2 ), q(p1 ), q(p2 )) the elements t ∈ T , q ∈ Q, and p1 6= p2 ∈ A are uniquely determi ...
