- #1

RJLiberator

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## Homework Statement

Let g(x) ∈ ℤ[x] have degree at least 2, and let p be a prime number such that:

(i) the leading coefficient of g(x) is not divisible by p.

(ii) every other coefficient of g(x) is divisible by p.

(iii) the constant term of g(x) is not divisible by p^2.

a) Show that if a ∈ ℤ such that [a]_p ≠ [0]_p, then [g(a)]_p ≠ [0]_p.

b) Show that if b ∉ ℤ such that

**_p = [0]_p, then [g(b)]_p^2 ≠ [0]_p^2.**

c) Conclude that g(x) has no roots in ℤ

Well, the problem I am having here is understanding the question at all.

So g(x) exists in Z[x].

What does Z[x]

It has a degree of at least 2, and we have p as a prime number.

We know the leading coefficient of g(x) is NOT divisible by p.

So this means we have C*x^n+...

Where c is not divisible by p.

Every other coefficient of g(x) is divisible by p.

So we have C*x^n + g*x^(n-1)+...

where g IS divisible by p.

The constant term fo g(x) is not divisible by p^2. So we now have

C*x^n + g*x^(n-1)+...+ z

where z is not divisible by p*p.

So we look at question a. If a exists in the integers such that [a]_p cannot equal [0]_p then [g(a)]p cannot equal [0]p.

So we have

C*a^2+g*a+z as an example polynomial.

and here is where I start to break down.

Can't decide what to do from here.

c) Conclude that g(x) has no roots in ℤ

## Homework Equations

## The Attempt at a Solution

Well, the problem I am having here is understanding the question at all.

So g(x) exists in Z[x].

What does Z[x]

*exactly*mean?It has a degree of at least 2, and we have p as a prime number.

We know the leading coefficient of g(x) is NOT divisible by p.

So this means we have C*x^n+...

Where c is not divisible by p.

Every other coefficient of g(x) is divisible by p.

So we have C*x^n + g*x^(n-1)+...

where g IS divisible by p.

The constant term fo g(x) is not divisible by p^2. So we now have

C*x^n + g*x^(n-1)+...+ z

where z is not divisible by p*p.

So we look at question a. If a exists in the integers such that [a]_p cannot equal [0]_p then [g(a)]p cannot equal [0]p.

So we have

C*a^2+g*a+z as an example polynomial.

and here is where I start to break down.

Can't decide what to do from here.