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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6310 • Algebra

Summer 2016 • Question 10

Question

Math 6310 • Algebra Summer 2016 • Question 10

Let \(p\) be a prime integer, and set \(f(x)=x^{p-1}+x^{p-2}+\cdots +x+1\). Suppose a prime integer \(q\) divides \(f(a)\) for some integer \(a\), prove that either \(q=p\) or \(q\equiv 1\bmod p\).

Use this to prove that the arithmetic sequence \(1,1+p,1+2p,\ldots \) contains infinitely many prime integers.

Raw LaTeX

Let \(p\) be a prime integer, and set
\(f(x)=x^{p-1}+x^{p-2}+\cdots+x+1\). Suppose a prime integer \(q\) divides \(f(a)\) for some integer
\(a\), prove that either \(q=p\) or \(q\equiv 1\bmod p\).

Use this to prove that the arithmetic sequence \(1,1+p,1+2p,\ldots\) contains infinitely many
prime integers.

Topic Tags

Cyclotomic Fields Exact Sequences of Modules