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Coolster7
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Homework Statement
Let f(x) = anxn + an-1xn-1 + ... + a1x + a0 be a polynomial where the coefficients an, an-1, ... , a1, a0 are integers.
Suppose a0 is a positive power of a prime number p.
Show that if [itex]\alpha[/itex] is an integer for which f( [itex]\alpha[/itex] ) = 0, [itex]\alpha[/itex] is also a power of p.
Homework Equations
The Attempt at a Solution
I substituted [itex]\alpha[/itex] into the equation in the place of x for each term. I also substituted in pn in the place of a0 as this is a positive power of a prime number p (as given in the question). This gave me:
f([itex]\alpha[/itex]) = an[itex]\alpha[/itex]n + an-1[itex]\alpha[/itex]n-1 + ... + a1[itex]\alpha[/itex] + pn = 0
I then decided to isolate pn by moving the other terms to the other side of the equation which gave me:
pn = -{an[itex]\alpha[/itex]n + an-1[itex]\alpha[/itex]n-1 + ... + a1[itex]\alpha[/itex]}
Is what I have done so far correct? I now have to show from this that [itex]\alpha[/itex] is also a power of p. I'm unsure what the next step is to do that.
Can anyone help please?
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