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QUESTION C. (a) Let k be a field and let n be a positive integer. Define what is meant by a monomial ideal in k[x,.....
Q9 6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a...
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
* (9) Let n be a positive integer. Define : Z → Zn by (k) =...
* (9) Let n be a positive integer. Define : Z → Zn by (k) = [k]. (a) Show that is a homomorphism. (b) Find Ker(6) and Im(). yrcises (c) To what familiar group is the quotient group Z/nZ isomorphic? Explain.
3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n)....
3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n). (b) Let P(n) be the statement E(n) nn1)." Complete the steps below to give a proof by induction that P(n) holds for every neZ+ i. Verify P(1) is true. (This is the base step.) ii. Let k be some positive integer. We assume P(k) is true. What exactly are we assuming is true? (This is the inductive hypothesis.) iii. What is the statement P(k...
Ok = (6) Let n be a positive integer. For every integer k, define the 2...
Ok = (6) Let n be a positive integer. For every integer k, define the 2 x 2 matrix cos(27k/n) - sin(2nk/n) sin(2tk/n) cos(27 k/n) (a) Prove that go = I, that ok + oe for 0 < k < l< n - 1, and that Ok = Okun for all integers k. (b) Let o = 01. Prove that ok ok for all integers k. (c) Prove that {1,0,0%,...,ON-1} is a finite abelian group of order n.
(i) Show that R is a subring of the polynomial ring Rx. | R{]4 (ii) Let k be a fixed positive integer and be the set of...
(i) Show that R is a subring of the polynomial ring Rx. | R{]4 (ii) Let k be a fixed positive integer and be the set of all polynomials of degree less than or equal to k. Is R[xk a subring of R[a]? 2r4+3x - 5 when it is (iii) Find the quotient q(x divided by P2(x) of the polynomial P1( and remainder r(x) - 2c + 1 in - (iv) List all the polynomials of degree 3 in Z...
subring of the polynomial ring R{z] (i Show that R is a (ii) Let k be a fixed positive integer and Rrk be the set of al...
subring of the polynomial ring R{z] (i Show that R is a (ii) Let k be a fixed positive integer and Rrk be the set of all polynomials of degree less than or subring of Ra (iii) Find the quotient q(x) and remainder r(x) of the polynomial P\(x) 2x in Z11] equal to k. Is Rr]k a T52r43 -5 when divided by P2(x) = iv) List all the polynomials of degree 3 in Z2[r].
subring of the polynomial ring R{z]...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we ...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
(a) Let n be any positive integer. Briefly explain (no formal proofs) why n > 1...
(a) Let n be any positive integer. Briefly explain (no formal proofs) why n > 1 ≡ ¬(n = 1). (b) Recall that a positive integer p is prime iff there do not exist a positive integers n and m, both greater than 1, such that p = nm. (I.e., Prime(p) means ¬∃n ∃m (n > 1 ∧ m > 1 ∧ p = nm).) Give a formal proof of the following: for any prime p, any positive integers n...
50. What is wrong with this "proof? "Theorem For every positive integer n = (n +...
50. What is wrong with this "proof? "Theorem For every positive integer n = (n + /2. Basis Step: The formula is true for n = 1. Inductive Step: Suppose that +Y/2. Then -(+972 +*+- +*+1)/2 + + + /- + 1). By the inductive hypothesis, we have + /2-[(++P/2, completing the + inductive step.
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
* (9) Let n be a positive integer. Define : Z → Zn by (k) = [k]. (a) Show that is a homomorphism. (b) Find Ker(6) and Im(). yrcises (c) To what familiar group is the quotient group Z/nZ isomorphic? Explain.
3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n). (b) Let P(n) be the statement E(n) nn1)." Complete the steps below to give a proof by induction that P(n) holds for every neZ+ i. Verify P(1) is true. (This is the base step.) ii. Let k be some positive integer. We assume P(k) is true. What exactly are we assuming is true? (This is the inductive hypothesis.) iii. What is the statement P(k...
Ok = (6) Let n be a positive integer. For every integer k, define the 2 x 2 matrix cos(27k/n) - sin(2nk/n) sin(2tk/n) cos(27 k/n) (a) Prove that go = I, that ok + oe for 0 < k < l< n - 1, and that Ok = Okun for all integers k. (b) Let o = 01. Prove that ok ok for all integers k. (c) Prove that {1,0,0%,...,ON-1} is a finite abelian group of order n.
(i) Show that R is a subring of the polynomial ring Rx. | R{]4 (ii) Let k be a fixed positive integer and be the set of all polynomials of degree less than or equal to k. Is R[xk a subring of R[a]? 2r4+3x - 5 when it is (iii) Find the quotient q(x divided by P2(x) of the polynomial P1( and remainder r(x) - 2c + 1 in - (iv) List all the polynomials of degree 3 in Z...
subring of the polynomial ring R{z] (i Show that R is a (ii) Let k be a fixed positive integer and Rrk be the set of all polynomials of degree less than or subring of Ra (iii) Find the quotient q(x) and remainder r(x) of the polynomial P\(x) 2x in Z11] equal to k. Is Rr]k a T52r43 -5 when divided by P2(x) = iv) List all the polynomials of degree 3 in Z2[r].
subring of the polynomial ring R{z]...
Let k 21 be a positive integer, and let r R be a non-zero real number. For any real number e, we would like to show that for all 0 SjSk-, the function satisfies the advancement operator equation (A -r)f0 (a) Show that this is true whenever J-0. You can use the fact that f(n) = crn satisfies (A-r)f = 0. (b) Suppose fm n) satisfies the equation when m s k-2 for every choice of c. Show that )...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
50. What is wrong with this "proof? "Theorem For every positive integer n = (n + /2. Basis Step: The formula is true for n = 1. Inductive Step: Suppose that +Y/2. Then -(+972 +*+- +*+1)/2 + + + /- + 1). By the inductive hypothesis, we have + /2-[(++P/2, completing the + inductive step.
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