Prove that if D is an integral domain, then either char(D) = 0 or char(D) = p for some prime p. In other words, prove that the characteristic of an integral domain cannot be a composite number

Q: 5. Let n Your-Reg-ID (7 digits number) and = let 21, 22, 23,..., Zn be the (distinct) n-th roots of…

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Q: Let R[r] be the set of all expressions 00 a = do+ajx+azx += i=0 where a¡ E R for all nonnegative…

A: We will be solving part (b) as mentioned. (b). Given ℝx be set of all expressions…

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: Show that x4 + 1 is reducible over Zp for every prime p.

A: In this question, we have to show that x4+1 is reducible over ℤp for every prime p.

Q: Let x be an integer. Prove that 2|x and 3|x if and only if 6|x. Generalize and prove.

A: According to the given information, let x be an integer. It is required to prove that:

Q: Suppose that there exist integers x and y such that ax + by = 1. Show that a and b are relatively…

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Q: Prove in an integral domain, a prime element is always irreducible.

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Q: Prove that x4 + 15x3 + 7 is irreducible over Q

A: In this question,  we have to prove that  x4 + 15x3 + 7 is irreducible over Q.

Q: The set of integers is not dense in cofinite topo log y of R True False O

A: The set of integers is not dense in co finite topology on ℝ.

Q: Suppose that a and b belong to an integral domain and b ≠ 0. Showthat <ab> is a proper subset…

A: Let R be an integral domain. A commutative ring R is said to be an integral domain if non zero…

Q: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

Q: 3x-4 *P 4

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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: irr(ꭤ,ℚ) and deg(ꭤ,ℚ

A: Given: 2+i To find:  The irr(ꭤ,ℚ) and deg(ꭤ,ℚ)for the given algebraic number ꭤ ∈ ℂ.

Q: Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

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Q: (7) The difference of two odd integers x and y is 4. Prove that x and y are relatively prime.

A: Given difference of two odd integers is 4.

Q: Let n and k be integers such that (n, k) = 1. Prove that (n, n − k) = 1.

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Q: Prove that gcd(a, b) is the least natural number representable in the form ax + by with x, y ∈ Z.

A: We have to show that gcd(a, b) is the least natural number representable in the form ax + by with x,…

Q: Prove that there are integers x and y such that 10x-13y=1

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

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Q: (4) Prove that there is no smallest positive real number.

A: I have to prove that there is no smallest positive real number . Suppose that there is a smallest…

Q: et a, b, c be non-zero integers. Suppose that x, y are integers such that ax + by = c. Show that for…

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Q: d/ Prove that the following numbers are real (for any 3 享

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Q: please send handwritten solution for part a

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Q: prove that for m ≥ 3 an integer, ordm(m − 1) = 2.

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Q: numbers, then show that Q(√2, √3)= Q(√2+√3)

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Q: Prove that 18000 + 28000 38000 Theorem (and not a calculator). 18000 + 58000 + +4° 8000 is a…

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Q: Find the minimal polynomial for 3√ 2 + 3√ 4 over Q.

A: Let x=23+43. To Find: Minimal polynomial for 23+43 over Q.

Q: Let a and b be integers which are not both zero, and let d= gcd(a, b). Prove that gcd (a/d,b/d)= 1.

A: Given that, a and b be integers and in both of them at least one is non-zero. And, greatest common…