irr(ꭤ,ℚ) and deg(ꭤ,ℚ

Q: Show that the following polynomials are irreducible over Q. (a) f(x) = 13x' + 15x* + 20x² + 21x + 19…

A: .

Q: Let a be a root of the polynomial x3+x+3, which is irreducible over Q. The polynomial x2-2 is also…

A: Given: a is a root of the polynomial x3+x+3, which is irreducible over ℚ. The polynomial x2-2 is…

Q: Let p > 2 be a prime number. If f(x) = 1' + x* + x² - x +1 € Z,[F] is divisible by the polynomial z…

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Q: 2. Let P(x) = ax² + bx + c be an irreducible polynomial in Q[x). Prove that (a) P(x) has two…

A: Let P(x)=ax2+bx+c be an irreducible polynomial in Q[x] a) To show P(x) has two distinct roots, let…

Q: Show that the set degree 1 polynomials with natural number coefficients, i.e.{nx+m | n, m∈N}, is…

A: A set S  is said to be countable if it has a one - to - one function as f:S→N.

Q: Now just as with Example 2.16, prove that if f, g, and three polynomials in R(r], then (f+ g) + h =…

A: Introduction: In mathematics, functions are the most fundamental aspect of calculus. Functions are…

Q: Find all irreducible factors of the polynomial x° – 1 over the field of rationals Q. Show that each…

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Q: Which of the following polynomials is irreducible over Q? * O x^4+x+1 3x 3-6x 2+x-2 O x*2+5x-6 ONone…

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Q: Let p be a prime and let q = p². Show that Z,[x] has a unit that is not a constant polynomial.

A: Given that, Let p be a prime and let q = p ^ 2 . Show that Zq[x] has a unit that is not a constant…

Q: Prove that ithe polynomial PCO =-2xP +2x2+loxx t6 has and fachor azero at M= 3, and fachm s ala the…

A: The given polynomial is px=-2x3+2x2+10x+6. It is given that x=3 is a root of the polynomial. If x=3…

Q: Find the number of ideals in Cx,y that contains both the polynomials y – (x – 1)³ +1 and p(x) in…

A: We have to find the number of ideals in ℂx,y that contains both the polynomials y-x-13+1 and px in…

Q: The number of irreducible polynomials over Z13 of the form x + ajr + a2 is 91 O 182 O 78 O 169

A: Given x2+a1x+a2 is a 2nd degree polynomial .For a 2nd degree polymonial , no of irraducible…

Q: Write a general polynomial q(z) of degree n with coefficients that are bo, b,, b2, ..b,. What are…

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Q: Consider the following polynomials in P3: a(x) = x³_2x²–2x+3 r(x) = 4x³–x²-3x+6 u(x) = 4x³+9x²+x+3…

A: I am going to solve the given problem by using some simple algebra to get the required result.

Q: Find ALL irreducible polynomials of degree 2 or 3 in Zr].

A: An irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the…

Q: The following polynomial 2 points below is: p (x) = a + 3x + 2 not irreducible polynomial over the…

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Q: 2. The Maclaurin polynomials for In(1 +x) are x2 Tn(x) = +... +(-1)n-1ª" x – 3 Use this to find an n…

A: The Maclaurin polynomial is an expansion of a function. This is used to estimate the value of the…

Q: If a polynomial is reducible over Q then it is reducible over Z. O True O False

A: To show whether the given sentence is true or false.

Q: Find the minimal polynomial for  ∛2 + ∛4  over Q.

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Q: (2) Find all the numbers z such that elz = 10i.

A: Consider the given. eiz=10i

Q: Determine the irreducible polynomial for 15 over each of the following (c) a = /3 fields (i). Q (ii)…

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Q: 18. Let xo < x, < x2, let {yo, y, y2} be given. Find the polynomial q(x) with deg(q) < 2 and such…

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Q: Compute the first 5 Legendre Polynomials, Po(x), P₁(x), -P₁(x). (These These are > Solutions, P,…

A: What is Legendre Polynomial: Legendre polynomial is a collection of orthogonal polynomials of degree…

Q: Prove that the polynomial f(x) 2an +a2", n € N, a E R \ {0} is not divisible by g(z) = r+a

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Q: Is the polynomial x2 – 4 in the ideal generated by the polynomials x + 22 – 4x – 4, x – 22 – 4x + 4,…

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Q: For any positive integer n, how many polynomials are there ofdegree n over Z2? How many distinct…

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Q: Take the digit in the tens place as b, the digit in the hundreds place as c, and the digit in the…

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Q: Let the chor a cteristic polynomial A, Then det (A)=? O 1 O o any real number. Does not exist.

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Q: List all the polynomials of degree 2 in Z2[x]. Which of these areequal as functions from Z2 to Z2?

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Q: Find the polynomial with thats has degree 3 and Zeros 2,-1 and 3

A: To find the polynomial with thats has degree 3 and Zeros 2, -1 and 3.

Q: rove or Disprove that the polynomial x⁴ − 2 ∈ Q[x] is solvable by radical over Q

A: Given, x4-2∈Qx. The equation x4-2 can be reduced to a cubic equation and as it is known that any…

Q: Given a positive integer d, define Q[vā] = {s(Va) : f € Q-]}. i.e. the set of real numbers that can…

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Q: Showthat S subring of RIKRZ+R3

A: For the solution of the problem follow the next step.

Q: Use Eisenstein's criterion to show that the polynomial x3 –- 3x? + 9x – 10 is irreducible over the…

A: Solution :-

Q: The number of irreducible polynomials over Z13 of the form x + ajx + az is O 91 182 78 O 169

A:

Q: Prove or disprove that there exist nonconstant polynomials in Zp[x] that are units if p is prime.

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Q: Which of the following polynomials in Z3 [x] is irreducible? (a) p (x) = x ^ 3 + x + 1 (b) p (x) =…

A: Useful result: If f(x)∈F[x] be a polynomial over a field F of degree two or three. Then f(x) is…

Q: (d) The cubic Legendre polynomial is P2(r) =(5x – 3r). Prove that it is orthog- onal (over [-1, 1)…

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Q: snip

A: Given{x,2x-x2,3x+2x2}To check dependentIf they are dependent , need to express  one function in…

Q: Find the standard inner product on P of the given polynomials. p= -2+ 4x + a2, q = 3+ 4x 3a2. (p, q)

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Q: Use the intermediate value theorem to show that the polynomial has a real zero between the given…

A: The given function is

Q: polynomial in V2+i and compute the conjugates of i over Express i as a Q(/2 + i).

A: As per the question we have to express i as a polynomial in (√2 + i) and then compute its conjugate…

Q: Letf(x) be a polynomial with integral coefficients. Iff(1) and f(2) both are odd integers, prove…

A: Let us assume that  f(x)=0 for some integer x=k, then x-k divides f(x). therefore  f(x)=(x-k)g(x) ;…

Q: Given two polynomials in from of x^3+x^2+x+c =0 x^3+x^2+x+c =0

A: Given: To find their intersection using matrices. Given two polynomials in from of…

Q: Let a be the positive real fourth root of 3. Factor the polynomial x4 – 3 into irreducible factors…

A: x4-3 =0x=31/4, -31/4,31/4i,-31/4i these all are irreducible with respect to Q…

Q: the legendre polynomial pn (x) has .a none of these .b n zeros of which only one is between – 1 and…

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Q: Find a third-degree polynomial that has integer coefficients and zeros and 3+i. Evaluate (V12 - V-3)…

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Q: Find the minimal polynomial satisfied by √2 + √5. over Q.

A: We have to find the minimal polynomial which satisfied by 2+5 over Q.

Q: Use Eisenstein's criterion to show that the polynomial x³ – 3x2 + 9x – 10 is irreducible over the…

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.