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Exercises 6.5 Exercises

1.

Let \(z = a + b \sqrt{3}\, i\) be in \({\mathbb Z}[ \sqrt{3}\, i]\text{.}\) If \(a^2 + 3 b^2 = 1\text{,}\) show that \(z\) must be a unit. Show that the only units of \({\mathbb Z}[ \sqrt{3}\, i ]\) are \(1\) and \(-1\text{.}\)

2.

The Gaussian integers, \({\mathbb Z}[i]\text{,}\) are a UFD. Factor each of the following elements in \({\mathbb Z}[i]\) into a product of irreducibles.
  1. \(\displaystyle 5\)
  2. \(\displaystyle 1 + 3i\)
  3. \(\displaystyle 6 + 8i\)
  4. \(\displaystyle 2\)

3.

Prove or disprove: If \(D\) is an integral domain, then every prime element in \(D\) is also irreducible in \(D\text{.}\)

4.

Prove that the field of fractions of the Gaussian integers, \({\mathbb Z}[i]\text{,}\) is
\begin{equation*} {\mathbb Q}(i) = \{ p + q i : p, q \in {\mathbb Q} \}. \end{equation*}

5.

Let \({\mathbb Z}[ \sqrt{2}i ] = \{ a + b \sqrt{2}i : a, b \in {\mathbb Z} \}\text{.}\)
  1. Prove that \({\mathbb Z}[ \sqrt{2} i ]\) is an integral domain.
  2. Find all of the units in \({\mathbb Z}[\sqrt{2} i]\text{.}\)
  3. Determine the field of fractions of \({\mathbb Z}[ \sqrt{2} i ]\text{.}\)
  4. Prove that \({\mathbb Z}[ \sqrt{2} i ]\) is a Euclidean domain under the Euclidean valuation \(\nu( a + b \sqrt{2}\, i) = a^2 + 2b^2\text{.}\)

6.

Consider the ring of integer polynomials \(\mathbb Z[x]\text{.}\)
  1. Prove that if \(d\) is a greatest common divisor of \(f = x\) and \(g = 2\text{,}\) then \(d\) is \(\pm 1\text{.}\)
  2. Prove that \(\pm 1\) cannot be written as \(sf + tg\) for any elements \(s\) and \(t\) in \(\mathbb Z[x]\text{.}\)
  3. Conclude that \(\mathbb Z[x]\) is not a Euclidean domain.

7.

Let \(D\) be an integral domain with a function \(\nu\) from \(D \setminus \{0\}\) to the set of non-negative integers, which satisfies the second condition of a Euclidean domain, but not necessarily the first. For any nonzero \(a\) in \(D\text{,}\) define \(\nu'(a)\) to be \(\min\{\nu(ab) : b \in D \setminus \{0\}\}\text{.}\) Show that \(D\) is a Euclidean domain with Euclidean valuation \(\nu'\text{.}\)

8.

Suppose that \(D\) is an integral domain and \(a\) and \(b\) are both nonzero elements of \(D\text{.}\) If \(a\) and \(b\) have a greatest common divisor, prove it is unique up to associates. That is, if \(d\) and \(d'\) are both greatest common divisors of \(a\) and \(b\text{,}\) then \(d\) and \(d'\) are associates.

9.

Let \(D\) be an integral domain. Define a relation on \(D\) by \(a \sim b\) if \(a\) and \(b\) are associates in \(D\text{.}\) Prove that \(\sim\) is an equivalence relation on \(D\text{.}\)

10.

Let \(D\) be a Euclidean domain with Euclidean valuation \(\nu\text{.}\) If \(u\) is a unit in \(D\text{,}\) show that \(\nu(u) = \nu(1)\text{.}\)

11.

Let \(D\) be a Euclidean domain with Euclidean valuation \(\nu\text{.}\) If \(a\) and \(b\) are associates in \(D\text{,}\) prove that \(\nu(a) = \nu(b)\text{.}\)

12.

Show that any non-zero, non-unit element \(a\) in a Euclidean domain \(D\) can be factored into irreducibles.

13.

Let \(D\) be a Euclidean domain with an irreducible element \(p\text{.}\) Suppose that \(a \in R\) is not divisible by \(p\text{.}\) Prove that \(pa\) is not square, i.e., there does not exist \(x \in D\) such that \(pa = x^2\text{.}\)

14.

Let \(D\) be a Euclidean domain and let \(F_D\) be its field of fractions.
  1. Show that any element in \(F_D\) can be written as
    \begin{equation*} u\frac{p_1\cdots p_n}{q_1 \cdots q_m}\text{,} \end{equation*}
    where \(u\) is a unit in \(D\) and \(p_1,\ldots, p_n, q_1, \ldots, q_m\) are distinguished irreducibles in \(D\text{,}\) such that no \(p_i\) is equal to any \(q_j\text{.}\)
  2. Suppose that \(d\) is an irreducible element in \(D\text{.}\) Show that there does not exist \(x\) in \(F_D\) such that \(x^2=d\text{.}\)

15.

For each of the following integers, is it possible to write it as a sum of two squares? If so, find an expression as a sum of two squares.
  1. \(\displaystyle 52 = 2^2 \cdot 13\)
  2. \(\displaystyle 343 = 7^3\)
  3. \(\displaystyle 3751 = 11^2 \cdot 31\)
  4. \(\displaystyle 7865=5 \cdot 11 \cdot 11 \cdot 13\)

16.

Find four different pairs of squares which sum to \(1105=5 \cdot 13 \cdot 17\text{.}\) Here, we don’t count simply reversing the order as a different sum of squares.

17.

Suppose that \(p\) is a prime and \(p \equiv 1 \pmod 4\text{.}\)
  1. Show that it is possible to write \(p^2\) as the sum of two positive squares.
  2. Show that the expression of \(p^2\) as the sum of two positive squares is essentially unique, meaning that if \(p^2 = a_1^2 + b_1^2 = a_2^2 + b_2^2\) then the set \(\{a_1^2, b_1^2\}\) is equal to \(\{a_2^2, b_2^2\}\text{.}\)

18.

Suppose that \(p\) is a prime and \(p \equiv 3 \pmod 4\text{.}\) Show that it is not possible to write \(p^2\) as the sum of two positive squares.

19.

Show that \(2x^2 -2x + 4\) is irreducible in \(\mathbb Q[x]\text{,}\) but not in \(\mathbb Z[x]\text{.}\)

20.

Show that \(7\) is irreducible in \(\mathbb Z[x]\text{,}\) but not in \(\mathbb Q[x]\text{.}\)

21.

Let \(\mathbb Z[x^2,xy,y^2]\) be the subset of \(\mathbb Z[x,y]\) consisting of polynomials
\begin{equation*} f=\sum_{i=0}^n \sum_{j=0}^m c_{ij}x^i y^j \end{equation*}
where \(c_{ij} = 0\) whenever \(i+j\) is odd.
  1. Prove that \(\mathbb Z[x^2,xy,y^2]\) is a subring of \(\mathbb Z[x,y]\text{.}\)
  2. What are all divisors of \(f=x^3y\) in \(\mathbb Z[x^2,xy,y^2]\text{?}\)
  3. What are all divisors of \(g=x^2y^2\) in \(\mathbb Z[x^2,xy,y^2]\text{?}\)
  4. Show that \(f\) and \(g\) do not have a greatest common divisor in \(\mathbb Z[x^2,xy,y^2]\text{.}\)

22.

Show that \({\mathbb Z}[\sqrt{5}\, i]\) is not a unique factorization domain.

23.

Prove or disprove: Every subdomain of a UFD is also a UFD.
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