Prove that the set of all polynomials whose coefficients are all evenis a prime ideal in Z[x].

Q: is a maximal ideal of Q[z].

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Q: Find all maximal ideals of Z_12:

A: Given ring is Z_12

Q: Show that a nonconstant polynomial from Z[x] that is irreducibleover Z is primitive.

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Q: Find all ideals of Z3 [i] . Show your work. (-

A: ℤ3i is clearly a field  because ℤpi is a field if p≡3 (mod 4). Thus 3≡3(mod 4) therefore ℤ3i is a…

Q: Let BCR be a subset of R. Prove that there is an ideal I of R such that B n I has at most one…

A: To show that ideals I and J  exist satisfying conditions 1) and 2)

Q: 17. Find a nontrivial proper ideal of Z x Z that is not prime.

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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: give an example of An ideal in Z that is not a prime ideal.

A: The objective is to write an example of an ideal in ℤ that is not a prime ideal.

Q: Find

A: We know that,  All ideals of Zn are kZn, Where k is divisor of n.

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: 8. a. Prove that I[x] = {do + ax + + a,x"|a, = 2k, for k, E Z), %3D %3! ... the set of all…

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Q: Show that the polynomial 1+x+x² +...+xP-1 . where p is a prime number, is irreducible over the field…

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Q: Find all maximal and prime ideals of Z10-

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Q: Let A and В be ideals in principal ideal domain R , where A + R and В + R . Prove that the ideal AB…

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Q: Let F be a field. Show that in F[x] a prime ideal is a maximal ideal.

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Q: Let ? be a commutative ring with 1 and ? be a proper ideal of ?. Prove that ? is prime if  and only…

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Q: There are . Polynomials of degree atmost n in the polynomial ring Zs (x]. O 5 + 5^n O 5^n O 5^(n+1)…

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Q: Prove that the characteristic of a field is either  or a prime.

A: We need to prove : The characteristic of a field is either 0 or a prime W.k.t if the field has…

Q: Let R be an integral domain. Prove that {0R} is a prime ideal.  Let R be a ring and let p ∈ R be…

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Q: show that an ideal A is maximal iff A is

A:

Q: 4. Prove that I[a] = {ao + a1x + ..+ ana" a; = 2k; for k; e Z}, the set of all polynomials with even…

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Q: Prove that f(x) is reducible over Z3. Write f(x) as a product of irreducible factors.

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Q: is a maximal ideal of Q[x).

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Q: Prove that the ideal <x> in Q[x] is maximal.

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Q: Suppose D is a PID. Prove that every non-zero prime ideal is maximal.

A: Suppose the R is a PID. We need to show that every non-zero prime ideal is maximal. In case you need…

Q: Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily…

A: Given: f(x) is a polynomial of degree n over field F. let n=1 and f(x)=ax+b here a,b=0. Roots of…

Q: Find the degree over ℚ of the splitting field over ℚ of the given polynomial in ℚ [x]. Given…

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Q: Find all monic irreducible polynomials of degree 2 over Z

A: A polynomial of degree 2 in Z3 [x] is irreducible if and only if it has no roots in Z3.

Q: Consider the ring of polynomials Q(z) , x²-1∈Q(z) Is aprinciple ideal ? Is a maximal ideal?

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Q: Find all values of a in Z7 such that the quotient ring Z7[x]/(p(x)) where p(x) = x³ + x² + ax + 3 is…

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Q: A proper ideal J of R is a maximal ideal if and only if R/J is a field.

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

A: Solution :-

Q: 2. List all the polynomials in Z3[x] that have degree 2.

A: To find all the polynomials in Z3[x] that have degree 2

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

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Q: For every positive integer n, there is a polynomial in Z[x] of degree n that is ireducible over Q.…

A: A simple application of Eisenstein's Criterion

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

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Q: Let R = Z[x] and let P = {f element of R | f(0) is an even integer}. Show that P is a prime ideal of…

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Q: The splitting field of an irreducible polynomial of degree n over F is a degree п.

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Q: There are Polynomials of degree atmost n in the polynomial ring Z3[x]. .... none 3 + 3^n 3^n 3^(n+1)

A: Option D

Q: The ring Z8[i] has no proper ideals True False

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Let S be the subring of Z[x] of all polynomials with even constant terms. Which of the statements…

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Q: There are ... Polynomials of degree atmost n in the polynomial ring Z3[x]. none 3^n O 3^(n+1) O 3 +…

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Q: [a Prove that I= { b. : a,b,c,d are even integers} is an ideal of M2(Z). d_ 1.

A: 1) The main aim is to show that I=acbd|a,b,c,d are even integersis an ideal in M2Z. to show that the…

Q: Let D be a principal ideal domain and let p E D. Prove that (p) is a maximal ideal in D if and only…

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Q: Prove that C is not the splitting field of any polynomial in Q[x].

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Q: Prove that every ideal in F[x], where F is a field, is a principal ideal

A: To show: Every ideal in F[x], where F is a field is a principle ideal

Q: 1. Let D be a quasilocal PID (that is, a PID with a unique maximal ideal). Prove that for all a, b e…

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Q: Prove directly that a maximal ideal is irreducible.

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Q: 12. Show that the set of all constant polynomials in Z[r] is a subring but not an ideal of Z[x]

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

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