Example 1 provided for reference.
Example 1 The polynomialx) X2+ X+1 is irreducible in Za[XI, since it has no roots in Z2. Thus (X)) is a maximal ideal in Z,[X), and Z[X]/(f(X is a field. Let us denote it by K. To see what K looks like, notice that the coset g(X) determined by any g(X)EZ2[X] is the same as that determined by some r(X) which is either 0 or has degree
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Let K= {0, 1,RX+1} be the four-element field constructed in Example 1 on 206-207. Write X2+X+ 1 a...
Problem 4. Consider the field Z2[x]/(F), where $ = x5 + x2 + 1. In this...
Problem 4. Consider the field Z2[x]/(F), where $ = x5 + x2 + 1. In this field, we write abcde as a notation for ax4 + bx3 + cx2 + dx +e, where a, b, c, d, e are elements of Z2. For example, 11010 is a notation for the element 1x4 + 1x3 + 0x2 + 1x+0 = x4 + x3 + x. Compute the following. Make sure to write all of your answers either as polynomials of degree...
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...
Let F49 be the field of 49 elements constructed in class. The definition of this field...
Let F49 be the field of 49 elements constructed in class. The definition of this field is F19={la(x)]F: a(r) e Z,a}} where Z7]is the ring of polynomials in r with coefficients in the field Z7 and a(x)p = {a(x)+ (1]zz + [4],)5(x) : 5(#) e Z7(a]} and addition is given by [a(r)]F+ [b(r)]F = [a(r) + b(2)]F and multiplication is given by [a(r)]F[b(x)]F = [a(z)b(1)]p. 1. Let Fa9t represent the ring of polynomials with coefficients in F9 (a) Show that...
Let p(x) = 24 + 23 +1€ Z2[2] and let a = [z] in the field...
Let p(x) = 24 + 23 +1€ Z2[2] and let a = [z] in the field E = Z2[z]/(p(x)), so a is a root of p(x). (a) (15 points) Write the following elements of E in the form aa+ba+ca+d, with a,b,c,d € Z2. i. a“, a, a6, and a 10 ii. a5 +a+ + a2 + 1 iii. (a? + 1)4 (b) (5 points) The set of units E* = E-{0} of the field E is a group of order...
=n. 1: (a) Let n = 2020. List all of the pairs (x, y) E Z2...
=n. 1: (a) Let n = 2020. List all of the pairs (x, y) E Z2 such that x2 - y2 (b) Let n = (39). Find the number of pairs (x,y) e Zé such that x2 + y2 = n2. (c) Write 650 as a product of irreducible elements in Z[i], then list all of the pairs (x, y) e Z2 with 0 < x <y such that x2 + y2 = 650.
Let F=Z_3, the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]....
Let F=Z_3, the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (ab)^3=(a^3)(b^3) The Remainder Theorem If a and b are in Z_3 then (a+b)^3=2^3+b^3 For all a in Z_3, a^3=a The first isomorphism...
Let F=Z_3 , the finite field with 3 elements. Let f(x) be an irreducible polynomial in...
Let F=Z_3 , the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (a+b)^3=a^3+b^3 If g is an automorphism of K leaves g(r) is a root of f(x) The Remainder Theorem The Factor Theorem...
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...
10. Camider the ring of plynicanials z,Ir, and let/ denote the elmmont r4 + 2a +...
10. Camider the ring of plynicanials z,Ir, and let/ denote the elmmont r4 + 2a + 1 a) (5 points) Show that the quotient rga)/ () is a field. b) (5 points) Let a denote the coset z()Regarding F as a vector space over Z2, find a basis for F coasisting of powers of a c) (5 poluts) How nuany elements dors F have? Justify your answer. d) (5 points) Compute the product afas t a) i.e. expand this product...
Problem 4. Consider the field Z2[x]/(F), where $ = x5 + x2 + 1. In this field, we write abcde as a notation for ax4 + bx3 + cx2 + dx +e, where a, b, c, d, e are elements of Z2. For example, 11010 is a notation for the element 1x4 + 1x3 + 0x2 + 1x+0 = x4 + x3 + x. Compute the following. Make sure to write all of your answers either as polynomials of degree...
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...
Let F49 be the field of 49 elements constructed in class. The definition of this field is F19={la(x)]F: a(r) e Z,a}} where Z7]is the ring of polynomials in r with coefficients in the field Z7 and a(x)p = {a(x)+ (1]zz + [4],)5(x) : 5(#) e Z7(a]} and addition is given by [a(r)]F+ [b(r)]F = [a(r) + b(2)]F and multiplication is given by [a(r)]F[b(x)]F = [a(z)b(1)]p. 1. Let Fa9t represent the ring of polynomials with coefficients in F9 (a) Show that...
Let p(x) = 24 + 23 +1€ Z2[2] and let a = [z] in the field E = Z2[z]/(p(x)), so a is a root of p(x). (a) (15 points) Write the following elements of E in the form aa+ba+ca+d, with a,b,c,d € Z2. i. a“, a, a6, and a 10 ii. a5 +a+ + a2 + 1 iii. (a? + 1)4 (b) (5 points) The set of units E* = E-{0} of the field E is a group of order...
=n. 1: (a) Let n = 2020. List all of the pairs (x, y) E Z2 such that x2 - y2 (b) Let n = (39). Find the number of pairs (x,y) e Zé such that x2 + y2 = n2. (c) Write 650 as a product of irreducible elements in Z[i], then list all of the pairs (x, y) e Z2 with 0 < x <y such that x2 + y2 = 650.
Let F=Z_3, the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (ab)^3=(a^3)(b^3) The Remainder Theorem If a and b are in Z_3 then (a+b)^3=2^3+b^3 For all a in Z_3, a^3=a The first isomorphism...
Let F=Z_3 , the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (a+b)^3=a^3+b^3 If g is an automorphism of K leaves g(r) is a root of f(x) The Remainder Theorem The Factor Theorem...
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...
10. Camider the ring of plynicanials z,Ir, and let/ denote the elmmont r4 + 2a + 1 a) (5 points) Show that the quotient rga)/ () is a field. b) (5 points) Let a denote the coset z()Regarding F as a vector space over Z2, find a basis for F coasisting of powers of a c) (5 poluts) How nuany elements dors F have? Justify your answer. d) (5 points) Compute the product afas t a) i.e. expand this product...
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