Let Rℤiabi With Ab In ℤ Be The Ring Of Gaussian Integers

Let Rℤiabi With Ab In ℤ Be The Ring Of Gaussian Integersa S

Let R=ℤ[i]={a+bi | a,b ∈ ℤ} be the ring of Gaussian integers. (a) Show that I={α(4−i) | α ∈ R} is an ideal of R. (b) Show that φ: R → ℤ/17ℤ, given by φ(a+bi)=a+4b, is a surjective ring homomorphism. (c) Now show that R/I is isomorphic with ℤ/17ℤ (as rings).

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The Gaussian integers ℤ[i], consisting of all complex numbers a+bi where a and b are integers, form a well-studied subring of the complex numbers. This ring exhibits properties similar to the ordinary integers, including ideals, homomorphisms, and quotient structures. The following analysis addresses the structure of a particular ideal within ℤ[i], the existence of a ring homomorphism onto a finite field, and the isomorphism between a quotient ring and this finite field.

(a) Demonstrating that I = {α(4 − i) | α ∈ R} is an Ideal of R

To show that I is an ideal in R, we need to verify two main properties: first, that I is a subring of R (closed under addition and additive inverses), and second, that for every r in R and every i in I, the product r·i is in I.

By construction, I is generated by the element 4 − i, meaning every element of I has the form α(4 − i) for some α in R. Since R is a ring, scalar multiplication by elements of R preserves closure, so I is closed under addition and additive inverses; for example, for any α, β in R, α(4−i) + β(4−i) = (α + β)(4−i), which is still in I. Similarly, for any α, the additive inverse is (−α)(4−i), also in I. This confirms that I is an additive subgroup of R.

Next, to verify absorption under multiplication by R, consider any r in R and i in I, with i = α(4−i). Then, r·i = r·α(4−i) = (rα)(4−i), which is in I since rα is in R. Therefore, I absorbs multiplication from R, satisfying the ideal criteria. Overall, the set I, generated by (4−i), forms a principal ideal of ℤ[i].

(b) Showing that φ: R → ℤ/17ℤ defined by φ(a+bi) = a + 4b is a surjective ring homomorphism

Define the map φ: R → ℤ/17ℤ by φ(a+bi) = a + 4b mod 17. To establish that φ is a ring homomorphism, we verify the properties of preservation of addition and multiplication:

• Additivity: For any a, b, c, d in ℤ,

φ((a+bi) + (c+di)) = φ((a + c) + (b + d)i) = (a + c) + 4(b + d) = a + 4b + c + 4d ≡ (a + 4b) + (c + 4d) mod 17.

Also, φ(a+bi) + φ(c+di) = (a + 4b) + (c + 4d) = a + c + 4(b + d), so additivity holds.

• Multiplicativity: For a+bi and c+di, their product is (ac−bd) + (ad + bc)i. Then,

φ((a+bi)(c+di)) = φ((ac−bd) + (ad + bc)i) = (ac−bd) + 4(ad + bc) ≡ ac−bd + 4ad + 4bc mod 17.

On the other hand, φ(a+bi)·φ(c+di) = (a + 4b)(c + 4d) = ac + 4ad + 4bc + 16bd. Since 16 ≡ -1 mod 17, this is equivalent to ac + 4ad + 4bc - bd mod 17.

Comparing both expressions confirms that φ preserves multiplication modulo 17, considering the relation 16 ≡ -1 in ℤ/17ℤ, thus verifying multiplicativity.

Surjectivity of φ is evident because for any element x in ℤ/17ℤ, we can choose a= x and b=0 in ℤ, then φ(a+0i)=a+0= x mod 17, covering all elements in the codomain.

(c) Showing that R/I is isomorphic with ℤ/17ℤ as rings

The key to establishing an isomorphism between R/I and ℤ/17ℤ lies in the ideal I and the homomorphism φ. Since I is generated by 4−i, and φ is a surjective ring homomorphism with kernel precisely equal to I, the First Isomorphism Theorem for rings applies.

To confirm this, we verify the kernel of φ:

• Suppose φ(a+bi) = 0 in ℤ/17ℤ; then a + 4b ≡ 0 mod 17, implying a ≡ -4b mod 17.
• Expressing a in terms of b, the element a+bi in the kernel is of the form (-4b) + bi = b(-4 + i).
• Therefore, every element in the kernel has the form b(-4 + i), which generates the ideal I. Since I = (4−i), and -4 + i differs by a unit from 4− i (both generate the same ideal, as units in ℤ[i]), the kernel of φ is precisely I.

Accordingly, by the First Isomorphism Theorem, R/ker(φ) ≅ ℤ/17ℤ, and since ker(φ)=I, we obtain R/I ≅ ℤ/17ℤ as rings. This demonstrates the aforementioned quotient is structurally identical to the finite field ℤ/17ℤ.

Conclusion

Through the analysis above, it has been shown that the set I generated by (4−i) is a principal ideal of the Gaussian integers; that the homomorphism φ defined by φ(a+bi) = a+4b mod 17 is a surjective ring homomorphism; and lastly, that the quotient ring R/I is isomorphic to ℤ/17ℤ, revealing a deep connection between the algebraic structures of Gaussian integers and finite fields. This framework demonstrates the rich interplay of ideals, homomorphisms, and quotient constructs within algebraic number theory, emphasizing the structure and symmetry within Gaussian integers.

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