Let p be a prime and d a divisor of p-1. show that the d th...

Let p be a prime and d a divisor of p-1. show that the d th powers form a subgroup of U(Z/pZ) of order (p-1)/d. Calculate this subgroup for p=11, d=5; p=17,d=4 ;p=19,d=6

Expert Solution

Colby Messinger answered 6 months ago

Let gcd(a, p) = 1 with p a prime. Show that if a has at least...

Let gcd(a, p) = 1 with p a prime. Show that if a has at least one square root, then a has exactly 2 roots. [hint: look at generators or use x^2 = y^2 (mod p) and use the fact that ab = 0 (mod p) the one of a or b must be 0(why?) ]

show that an integer n > 4, is prime iff it is not a divisor of...

show that an integer n > 4, is prime iff it is not a divisor of (n-1)!

11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]p, [2]p, ....

11.4 Let p be a prime. Let S = ℤ/p - {0} = {[1]p, [2]p, . . . , [p-1]p}. Prove that for y ≠ 0, Ly restricts to a bijective map Ly|s : S → S. 11.5 Prove Fermat's Little Theorem

Let a and b be positive integers, and let d be their greatest common divisor. Prove...

Let a and b be positive integers, and let d be their greatest common divisor. Prove that there are infinitely many integers x and y such that ax+by = d. Next, given one particular solution x0 and y0 of this equation, show how to find all the solutions.

Let p be an integer other than 0, ±1. (a) Prove that p is prime if...

Let p be an integer other than 0, ±1. (a) Prove that p is prime if and only if it has the property that whenever r and s are integers such that p = rs, then either r = ±1 or s = ±1. (b) Prove that p is prime if and only if it has the property that whenever b and c are integers such that p | bc, then either p | b or p | c.

7. Let E be a ﬁnite extension of the ﬁeld F of prime characteristic p. Show...

7. Let E be a ﬁnite extension of the ﬁeld F of prime characteristic p. Show that the extension is separable if and only if E = F(Ep).

How to show that there are infinitely many prime p of the form p= 1+5k or...

How to show that there are infinitely many prime p of the form p= 1+5k or p=4+5k

1. Consider the group Zp for a prime p with multiplication multiplication mod p). Show that...

1. Consider the group Zp for a prime p with multiplication multiplication mod p). Show that (p − 1)2 = 1 (mod p) 2. Is the above true for any number (not necessarily prime)? 3. Show that the equation a 2 − 1 = 0, has only two solutions mod p. 4. Consider (p − 1)!. Show that (p − 1)! = −1 (mod p) Remark: Think about what are the values of inverses of 1, 2, . . ....

(a) Let G be a finite abelian group and p prime with p | | G...

(a) Let G be a finite abelian group and p prime with p | | G |. Show that there is only one p - Sylow subgroup of G. b) Find all p - Sylow subgroups of (Z2500, +)

2. (a) Let p be a prime. Determine the number of elements of order p in...

2. (a) Let p be a prime. Determine the number of elements of order p in Zp^2 ⊕ Zp^2 . (b) Determine the number of subgroups of of Zp^2 ⊕ Zp^2 which are isomorphic to Zp^2 .

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