Abstract algebra proof involving prime numbers

[christinamora]

The question states prove,
If p is prime and p | a^n then p^n | a^n

I am pretty sure I have i just may need someone to help clean it up.

There are two relevant theorems i have for this.
the first says p is prime if and if p has the property that if p | ab then p | a or p | b

the second one is that if p is prime and p | a1a2a3...an, then p must divide one of the a_i.

so for the proof i am assuming p | a^n which i can rewrite as

p | a*a*a...an-1*an. so this is saying p(q) = a*a*a...an-1*an for some integer q.

now if I look at p^n | a^n that's the same as

p*p*p...pn-1*pn | a*a*a...an-1*an

well p(p*p*p...pn-1*p) | a*a*a...an-1*an

is that the way to go?
Or maybe before when i had that p(q) = a*a*a...an-1*an for some integer q.

just set q = p^n-1 so that p(q) = p^n.

I feel like the later way should do it.
Is this right?

 

[Dick]

It's actually hard to know whether you have it or not. Because that's pretty unreadable. If you know that "if p is prime and p | a1a2a3...an then p must divide one of the a_i" and you apply that to p | a^n=a*a*a*...a (n times), what do you conclude about the divisibility of a by p? It's just the special case where all of the a_i are equal to a.

 

[christinamora]

ok so it is just a special case, so if p | a^n then the fact that p should divide one of the a_i means simply p|a, since every a_i is a.

so from p |a^n implies p |a .

so if p divides a we have that p(q) = a for some integer q.

since we have an equation, i can raise both sides to the n power,
so now i have p^n(q^n) = a^n which implies p^n | a^n.

that looks like it should be good right?

 

[Dick]

ok so it is just a special case, so if p | a^n then the fact that p should divide one of the a_i means simply p|a, since every a_i is a.

so from p |a^n implies p |a .

so if p divides a we have that p(q) = a for some integer q.

since we have an equation, i can raise both sides to the n power,
so now i have p^n(q^n) = a^n which implies p^n | a^n.

that looks like it should be good right?

Now that looks right.