← 2013 Paper 2

UPSC 2013 Maths Optional Paper 2 Q3a — Step-by-Step Solution

15 marks · Section A

Euclidean domains · Algebra · asked 5× in 13 yrs · Read the full method →

Question

Let $J=\{a+bi\mid a,b\in\mathbb Z\}$ be the ring of Gaussian integers (subring of $\mathbb C$). Which of the following is $J$: Euclidean domain, principal ideal domain, unique factorization domain? Justify your answer.

Technique

Standard verification of the Euclidean norm $N(a+bi)=a^{2}+b^{2}$ with the division algorithm via rounding to nearest integer.

Solution

Claim: $J=\mathbb Z[i]$ is all three — Euclidean domain, principal ideal domain, and unique factorisation domain.

Strategy. Prove $J$ is Euclidean (the strongest of the three). The implications Euclidean $\Rightarrow$ PID $\Rightarrow$ UFD are standard, so the other two follow.

Step 1 — $J$ is a Euclidean domain

Define the norm

This is non-negative integer-valued, with $N(\alpha\beta)=N(\alpha)N(\beta)$ (multiplicativity), and $N(\alpha)=0\iff\alpha=0$.

Division algorithm: for $\alpha\in J,\;\beta\in J\setminus\{0\}$, find $\gamma,\rho\in J$ with $\alpha=\gamma\beta+\rho$ and $N(\rho)<N(\beta)$.

Proof. Compute the rational quotient $\alpha/\beta=p+qi\in\mathbb Q(i)$ (well-defined since $\beta\ne 0$). Choose integers $m,n$ with $|m-p|\le 1/2$ and $|n-q|\le 1/2$ (round to nearest integer). Set $\gamma=m+ni\in J$. Then

Take norms:

So $N$ is a Euclidean function on $J$, and $J$ is a Euclidean domain.

Step 2 — Euclidean $\Rightarrow$ PID

Standard theorem. Sketch. For any non-zero ideal $I\subset J$, pick $\beta\in I$ minimising $N(\beta)$. For any $\alpha\in I$, division gives $\alpha=\gamma\beta+\rho$ with $N(\rho)<N(\beta)$. But $\rho=\alpha-\gamma\beta\in I$, and by minimality of $N(\beta)$, $\rho=0$. So $\alpha=\gamma\beta\in(\beta)$. Hence $I=(\beta)$ — principal.

Step 3 — PID $\Rightarrow$ UFD

Standard theorem. Sketch. In a PID, every irreducible is prime (since irreducibles generate maximal ideals = prime ideals). Combined with the ascending-chain condition (Noetherian, automatic in PIDs), every non-zero non-unit has a factorisation into irreducibles, and the factorisation is unique up to units and reordering.

Conclusion

Answer