For an irreducible polynomial p(X) = 1 + X + X³ € F₂[X] consider F8 generated by p(X). Fill the blanks so that the matrix is a generator matrix of CRS (8,3) over Fg. [100] [100] [100] [100] [100] [100] [100] [100] [010] [110] (a) [101] [01: 001] [101] [011] [111] (b) [111] [000] [100] [110] [000] [100]

Q: Let p be a prime. Are there any nonconstant polynomials in Zp[x]that have multiplicative inverses?…

A: Given: A prime number p. To check: Whether any nonconstant polynomials in Zp[x] have multiplicative…

Q: Calculate GCD(x3 + 2x2 + 3x , x2 + 3x + 2) in Z5 [x].

A: We have to find GCD x3+2x2+3x,x2+3x+2 We will use Euclidean Algorithm Let…

Q: The multiplicative inverse of 1+3x in Z,[x] is: О бх+1 O 1/(3x+1) None

A: Inverse property: a*a-1=e In Polynomials    A non-constant polynomial can never have an inverse…

Q: Let A= 4 4 2 1-1 (a) Compute det(A) using the cofactor expansion along the third row. (b) Is A…

A: The adjoint of a square matrix A = [aij]n×n is defined as the transpose of the matrix [Aij]n×n ,…

Q: g(x) = [[ ½ x ]] %3D

A: Given : g(x)=12x

Q: Given g(z)- In(8z +z), the derivative g(z) is %3D O 8z + O 9z O 72z +1 7278+1 O 72r +1

A:

Q: Find the splitting field ofx4 + x2 + 1 = (x2 + x + 1)(x2 - x + 1)over Q.

A: The extension field K of a field F is called splitting field for the polynomial fx∈Fx if fx factors…

Q: Determine the first 3 terms in the Maclaurin expansion of y = esin², X + X + x².

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Q: Example 2 Compute (a) (1+i)/(3-i) and (b) (1+i)4.

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Q: Let f(x) = x² +1 € Z3[x] and let R= Z3[x]/I, where I = (f(x)). (a) Show that R is a field with 9…

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Q: (b) The characteristic polynomial of A is (A² + 4 +13)²(x - 5)². find all possible Jordan canonical…

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Q: Show that the function defines an inner product for polynomials p(x) = a0 + a1x + . . . + anxn and…

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Q: Let z E (-1,0) and let R, (z) be the remainder in Maclaurin's formula for In(1+z) in Cauchy's form.…

A: Given the function is fx=ln1+x

Q: Let h(x) = -3) Then, h(x) has a zero on (0, o). —1) (ӕ+2)(2+3а) O True False

A:

Q: If Z1 = 4 + 5i and Z 2 = -1 + I,  express in the form a + bi  (i) Z1 + Z2   (ii) Z1 – Z2

A: Formula used  sum of two complex number is given by  (a1+b1i)+(a2+b2i)=(a1+a2)+i(b1+b2)

Q: x² Vx² + 1 .2 Expand In in terms of In x, In(x² + 1) and In(x+ 1). (x+ 1)7

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Q: Show that the generating function for compositions into an even number of parts, each of which is…

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Q: Find an algebraic integer a in a Trace(a) = 17. quadratic field with N(a) 31 and

A: To find: An algebraic integer α in a quadratic field with Nα=31 and Traceα=17.

Q: Let α be a zero of x3 + x2 + 1 in some extension field of Z2. Solve the equation (α + 1) x + α = α2…

A: Given that α be a zero of x3+x2+1 in some extension field of ℤ2. We have to solve the equation…

Q: If A is an adjoint operator on H=K2 and  for x=(x(1), x(2)), we define A(x)=(x(1)-x(2), x(1)+x(2)),…

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Q: Prove that if f(n) is multiplicative, then so is f(n)/n.

A: We have to prove that if f(n) is multiplicative, then f(n)n is also a multiplicative function.

Q: A = [ 1   0   0   0 ]       [ -4  1   0   0 ]        [ 1  -1   1   0 ]       [ 7  -2   1    1 ]

A:

Q: Find the splitting field ofx4 + x2 + 1 =  (x2 + x + 1)(x2 - x + 1)over Q.

A: Given: x4+x2+1=x2+x+1x2-x+1To find: The splitting field of x4+x2+1=x2+x+1x2-x+1 over Q.

Q: Which pairs of polynomials f, 9 € C[X] do have exactly one common root? O f- (X - 1), g = (X + X2+X+…

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Q: Find the wronskian of f1= x2-6x, f2=x2+6x, f3= x2-3

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Q: If [a] = [1] in Z_n, prove that gcd(a, n) = 1. Show by example that the converse may be false.

A: The given data is: a=1 in Zn To prove gcd(a, n)=1

Q: Let f(x) be a fixed integer polynomial, and let m be a fixed positive integer. Denote the number of…

A: The details are provided in step 2.

Q: P"n+1 (x) – xP", (x) = (n+1)P,(x) = P,(x) %3D -

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Q: The multiplicative inverse of x+ x+1 in Z2[x]/ is: X^4 x^3 None X+1

A: