If p(x) is a polynomial in Zp[x] with no multiple zeros, show thatp(x) divides xpn - x for some n.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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If p(x) is a polynomial in Zp[x] with no multiple zeros, show that
p(x) divides xpn - x for some n.

Expert Solution
Step 1

Given that px is a polynomial in Zpx with no multiple zeros.

We have to show that px divides xpnx for some n.

Let E be a splitting field of px.

Let a1,a2,a3,.....,amE be the set of zeroes of px.

Since, px does not have multiple root, hence it can be seen that the elements of a1,a2,a3,.....,am are distinct.

Also, px=i=1mxai.

The splitting field of px is a finite extension of Zp.

It follows that EGFpn for some nN.

We know that, Every elements of GFpn is a zeros of xpnx.

It follows that ai is a zeroes of xpnx for any 1im.

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