Question

An element [a] of Zn is said to be idempotent if [a]^2 = [a]. Prove that if p is a prime number, then [0] and [1] are the only idempotents in Zp. (abstract algebra)

Answer #1

An element [a] of Zn is said to be idempotent if [a]^2 = [a].
Prove that if p is a prime number, then [0] and [1] are the only
idempotents in Zp. (abstract algebra)

Prove that every nonzero element of Zn is either a unit or a
zero divisor, but not both.

Let p and q be primes. Prove that pq + 1 is a square if and only
if p and q are twin primes. (Recall p and q are twin primes if p
and q are primes and q = p + 2.) (abstract algebra)

4. Prove that if p is a prime number greater than 3, then p is
of the form 3k + 1 or 3k + 2.
5. Prove that if p is a prime number, then n √p is irrational
for every integer n ≥ 2.
6. Prove or disprove that 3 is the only prime number of the form
n2 −1.
7. Prove that if a is a positive integer of the form 3n+2, then
at least one prime divisor...

Let Zn = {0, 1, 2, . . . , n − 1}, let · represent
multiplication (mod n), and let a ∈ Zn. Prove that there exists b ∈
Zn such that a · b = 1 if and only if gcd(a, n) = 1.

Problem 2: (i) Let a be an integer. Prove that 2|a if and only
if 2|a3.
(ii) Prove that 3√2 (cube root) is irrational.
Problem 3: Let p and q be prime numbers.
(i) Prove by contradiction that if p+q is prime, then p = 2 or q
= 2
(ii) Prove using the method of subsection 2.2.3 in our book that
if
p+q is prime, then p = 2 or q = 2
Proposition 2.2.3. For all n ∈...

Abstract algebra
Either prove or disprove the following statements. Be sure to
state needed theorems or supporting arguments.
Let F ⊂ G be finite fields. Then F, G have the same
characteristic,say p. Moreover, if p > 0 then logp (|F|) divides
logp (|G|).

Prove or disprove the following statement: 2^(n+k) is an element
of O(2^n) for all constant integer values of k>0.

(b) If n is an arbitrary element of Z, prove directly that n is
even iff n + 1 is odd. iff is read as “if and only if”

Prove or disprove (a) Z[x]/(x^2 + 1), (b) Z[x]/(x^2 - 1) is an
Integral domain.
By showing (a) x^2+1 is a prime ideal or showing x^2 + 1 is not
prime ideal.
By showing (b) x^2-1 is a prime ideal or showing x^2 - 1 is not
prime ideal.
(Hint: R/I is an integral domain if and only if I is a prime
ideal.)

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