Let F be a field and let f(x), g(x), n(x). k(x), and h(x) be polynomials in F[x]. Prove that if f(x) = g(x)(mod n(x)) and h(x) = k(x)(mod n(x)) where degree of n(x) > 0, then f(x) + h(x) = (g(x) + k(x))(mod n(x)).

Q: Show that f(x) = x6 − 2ax³ + a is inseparable over Z3 (a), with a in any extension field of Z₂l

A: Given that                       f(x)=x6-2ax3+a We have  to show that f(x) is inseparable over Z3(a)…

Q: Let R be an integral domain and let a, b, c € R. Suppose that ab = ac with a 40. Prove that b = c._

A: Since, the image contains two different problems, according to the guidelines, I am solving the…

Q: 1. Factorize each of the following polynomials over Z5 [x]. a) f(x) = x³ + 3x² + 3x + 2 b) f(x) = x¹…

A: The important thing to observe is that we have to check polynomials modulo 5 while factoring over…

Q: Show that: f(x) = x² + x + 2 € Z₂[x] is irreducible. Further, find the order of f(x).

A: Irreducible polynomial and it's order

Q: How many ways are there to draw 3 winners out of 200 people, if first Prize is $2000, 2nd prize is…

A: We have to find correct option by finding number of ways to draw three winners out of 200 satisfying…

Q: For an irreducible polynomial p(X) = 1+X + X² € F[X] consider F4 generated by p(X). Find x1, x2 E F,…

A: Sol

Q: 2. Let R be a U.F.D. and f (x), g (x) e R [x]. If f(x) g (x) is a primitive polynomial, then f (x)…

A:

Q: Let α be a zero of f (x) = x2 + 2x + 2 in some extension field of Z3.Find the other zero of f (x) in…

A: The given polynomial is f(x)=x2+2x+2 and α is one of the roots. To Find: the another root of the…

Q: let f(z) be polynomial of degree at least two. show that |f(z)|>|z| for large |Z|.

A: We prove this result by contradiction.

Q: IfF is a field of charact f(x) = x²*

A:

Q: Let F be a field and let p(x) E F[x]. If f(x), g(x) E F[x] and deg f(x) < deg p(x) and deg g(x) <…

A: Given: f(x)∈F[x], g(x)∈F[x], p(x)∈F[x] such that  degf(x)&lt;deg(p(x)) and degg(x)&lt;deg(p(x)), and…

Q: Let g(x) and h(x) belong to Z[x] and let h(x) be monic. If h(x) divides g(x) in Q[x], show that h(x)…

A: Given: Let's take g(x) and h(x) belongs to Z(x) and take h(x) as monic.

Q: Let & be a zero of f(x)=x2 +2x+2 in some extension field of Z3. Find the other zero of f (x) in…

A: Consider the given function fx=x2+2x+2. Note that, as the given function is quadratic, the function…

Q: Let the p.d.f. f(x) be positive on and only on the nonnegative integers .Give that f(x)=4/x f(x-1),…

A: Given that:

Q: Let f(x) = axm + am-prm-1 + . . . + a, and g(x) = b,x"+ b,-px"-1 + ... + bo belong to Q[x] and…

A: Given that, f(x)=amxm+am-1xm+....+a0∈Q[x]g(x)=bnxn+bn-1xn-1+....+b0∈Q[x] This implies that ai,bi∈Q.…

Q: Let x0, x1, ..., Xn be distinct points on the interval [a, b], where Xo <x₂ < x₂ <--- < x₂ and let f…

A:

Q: Let f(x)= x3 + x2 + x +1  ∈ Z2[x]. Write f(x) as a product of irreduciblepolynomials over Z2.

A: Given polynomial is: f(x)=x3+x2+x+1 Since, f(1)=0 So, x-1 is a factor. Now, factorize the…

Q: Let f(x) in F[x] be a nonconstant polynomial. Then the following statements are equivale

A:

Q: Problem 2.1 Suppose that a number z is to be selected from the real line R, and let A, B, and C be…

A: Given A=x:1≤x≤5B=x:3≤x≤7C=x:x≤0 Solution a: Let U=R be the universal set A={x: 1≤x≤5}=x∈[1, 5] So…

Q: Let K be an extension of a field F and f(x) be a polynomial of positive degree over F, then alpha∈K…

A: be an extension of a field and is a polynomial of positive degree over .We have to prove that  is a…

Q: If (2) and (2) are monic polynomials of minimal degree for which A is a root then /(a)=g(a). True…

A:

Q: 3. Suppose f(x) and g(x) are monic integer polynomials. Prove that f(x)|g(x) in Q[x] if and only if…

A: Integer polynomials are polynomials having integer coefficients. And the monic polynomials have…

Q: Let f (x), g(x) ∈ F [x]. (a) Show that if f (x)|g(x) and g(x)|f (x), then there is some c ∈ F so…

A: If f(x)|g(x) and g(x)|f(x) then by division algorithm there must exist polynomials.

Q: ) Prove that, if a and b are nonnegative integers such that a =4 b, then 3ª =5 3b. Using (a),…

A:

Q: (a) Prove that, if a and b are nonnegative integers such that a = 4 b, then 3ª =5 3b. (b) Using (a),…

A: (a) Let a ≡4 b          =&gt; 4 | (a - b)          =&gt; a - b = 4k for some integer k.…

Q: Let F be a field, let n be a positive integer, and let c(x) E F[x] be a polynomial of degree n. Let…

A:

Q: Show that over the ring Z/30Z[x] the polynomial f(x)=x has the factorization…

A:

Q: Let fx)=x+x +2 be a Polynomial in Z3[x]. Which of the following statements is True O f(x) is…

A: The given polynomial function is fx=x3+x2+2 in ℤ3x. Substitute x=0, 1 and 2 in fx=x3+x2+2 and…

Q: Let I/Q be an extension of fields and f(X) = K[X] is an irreducible polynomial such that the degree…

A: Given:  LQ as an extension of fields and fX∈KX is an irreducible polynomial such that the degree of…

Q: Let g(x) be a polynomial in Z₂[x]. Prove that if the polynomial code C generated by g(x) with length…

A:

Q: 2 (a) Prove that, if a and b are nonnegative integers such that a 4 b, then 3ª = 3. (b) Using (a),…

A: Given a, b are non negative integers such that a≡4b . Then to show 3a≡53b. To show the function…

Q: If f (2) and g(2) are monic polynomials of minimal degree for which A is a root then f(2)=g(2).

A: Given that f(λ) and g(λ) be the monic polynomials. the degree is minimum. and  given that the root…

Q: f(x) = x² − 3 is the minimal polynomial of √3 over Q,

A: We have that              f(x) = x2-3  be a polynomial of two degree.

Q: 3. Let F be a field and let a(x), b(x), c(x), d(x), and e(x) be polynomials in F[x] with a(x) ‡ 0F.…

A: To show that e(x) divides a(x)d(x), we need to prove that there exists a polynomial q(x) such that…