Exercises 5.4 Exercises
2.
Show that \(\Z[\sqrt{2}]=\{a+b\sqrt{2}:a,b\in\Z\}\) is ED.3.
Suppose that \(A\) is a PID and it is a subring of an integral domain \(B\text{.}\) Show that for \(a_1,a_2\in A\) if \(d=\gcd(a_1,a_2)\in A\) then \(d=\gcd(a_1,a_2)\in B\text{.}\)4.
Show that the map \(\ev\colon\Z[x]\to\Z[\sqrt{-5}]\) given by \(f(x)\mapsto f(\sqrt{-5})\) is a ring epimorphism. Show that \(\Z[\sqrt{-5}]\) is not a UFD. We prove in the next chapter that \(\Z[x]\) is a UFD. This shows that epimorphism of a UFD need not be a UFD.
Hint.
\(9=3\cdot 3=(2+\sqrt{-5})(2-\sqrt{-5})\in\Z[\sqrt{-5}]\text{.}\)
5.
Show that if \(A\) is an integral domain but not a field then \(A[x]\) is not a PID.
Hint.
If \(A[x]\) is a PID then the ideal \((x)\) is maximal. Indeed, if \((x)\subset I\) then \(I=(d)\) (since \(A\) is a PID), and \(x=de\) for some \(e\in A[x]\text{.}\) Degree consideration implies that \(x\sim d\) or \(d\sim 1\text{,}\) i.e., either \(I=(x)\) or \(I=A[x]\text{.}\) Hence \((x)\) is maximal and thus \(A[x]/(x)\simeq A\) is a field.
6.
Show that if \(\varphi\colon A\to B\) is a ring isomorphism, and \(A\) is a UFD then \(B\) is a UFD.7.
Determine irreducible elements of \(\Z[i]\text{.}\)8.
Let \(p\equiv 1\mod 4\) be a prime number. Show that there exists \(n\in\Z\) such that \(p\mid (n^2+1)\text{,}\) i.e., \(n^2\equiv -1\mod p\text{.}\)
Hint.
Consider \(n=1\cdot 2\cdots\frac{p-1}{2}\) and use Wilson’s theorem, i.e., \((p-1)!\equiv -1\mod p\text{.}\)
9.
Suppose that \(p\equiv 1\mod 4\) is a prime number. Show that \(p\) is reducible in \(\Z[i]\text{.}\)
Hint.
By the above exercise there exists \(n\in\Z\) such that \(p\mid (n^2+1)\text{.}\) In \(\Z[i]\text{,}\) \(n^2+1=(n+i)(n-i)\text{.}\) Assume that \(p\) is irreducible in \(\Z[i]\text{.}\) Thus \(p\) is a prime element in \(\Z[i]\) (why?). This implies that either \(p\mid n+i\) or \(p\mid n-i\text{.}\)
10.
If \(p\in\N\) is a prime number such that \(p\equiv -1\mod 4\) then \(p\) is a sum of two squares in \(\Z\text{.}\)
Hint.
Using the above exercise write \(p=xy\) with \(x,y\in\Z[i]\) such that \(y\) is irreducible in \(\Z[i]\) and \(x\) is a non-unit in \(\Z[i]\text{.}\) Taking Euclidean norm defined on \(\Z[i]\text{,}\) and using the fact that \(\Z\) is a UFD we get the result.
