#19 in Abstract algebra books
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Reddit mentions of Galois Theory (Universitext)
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I found this in Galois Theory by Rotman
Rather than try to explain the proof here, I scanned in three pages from the book - I think this is allowed under the fair use clause of the copyright act. You can find the relevant pages here.
The pages are numbers 43, 66 and 67.
Page 43 has two exercises - the first and last - that give you hints on how to prove x^4 - 10x^2 +1 is irreducible over Q. (I don't think this part is difficult.)
The next two pages give a proof due to Gerald Janusz of the actual reducibility over Zp. The proof is not conceptually difficult but it is, in my not very humble opinion, messy.
Aleph_not is correct.
In the end, the original polynomial factors into two quadratics in Zp.
The hen-scratching on the outside is mine.
I hope this helps.
Galois Theory