Question

Describe the set of primes p for which x^2-11 splits into linear factors over Z_p

03 Feb 2024
Answer :
Word Count : 477

To describe the set of primes \( p \) for which the polynomial \( x^2 - 11 \) splits into linear factors over \( \mathbb{Z}_p \) (the field of integers modulo \( p \)), we need to find the primes \( p \) for which \( x^2 - 11 \) has distinct roots modulo \( p \).

First, let's analyze \( x^2 - 11 \) over the field \( \mathbb{Z}_p \). By the Fundamental Theorem of Algebra, a polynomial of degree \( n \) has at most \( n \) roots. So, if \( x^2 __________ _________ ______ _________ ___.
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