(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

1) Show that the kernel of the homomorphism [itex] \theta: \mathbb{Z}\rightarrow \mathbb{Z}_{10} [/itex] defined by [itex] \theta(a+bi) = [a+3b]_{10}, a,b \in \mathbb{Z} [/itex] is [itex] <1+3i> [/itex] (i.e. the ideal generated by 1+3i).

3. The attempt at a solution

My answer confuses me. It shows that any element of [itex] <1+3i> [/itex] is indeed in the kernel (one way proof), then to show the other way it says;

Conversely let [itex] a+bi \in ker(\theta) \Rightarrow a+3b \equiv 0mod10 \Rightarrow 3a \equiv bmod10 [/itex]. Therefore [itex] a+bi \in <1+3i> [/itex].

I'm not sure how it concludes the "therefore" part so easily. For example, [itex] 4+2i [/itex] would be in the kernel, because [itex] 12 \equiv 2mod10 [/itex] but from this proof it's not intuitively obvious to me that that [itex] 4+2i [/itex] is contained in [itex] <1+3i> [/itex]. It is though, because [itex] (1+3i)(1-i) = 4+2i [/itex]. But exactly how does the latter half of the proof show that, (just as an example), [itex] 4+2i [/itex] is contained within [itex] <1+3i> [/itex]?

Basically, how does [itex]3a \equiv bmod10 \Rightarrow a+bi \in <1+3i> [/itex]?

I've found a way to prove this, but it requires a bit of work;

An element is in the kernel if

[itex] 3a - b = 10k [/itex]

[itex] b = 3a - 10k [/itex]

So any complex number of the form;

[itex] a + (3a-10k)i [/itex]

Is in the kernel.

Is [itex] a + (3a-10k)i \subseteq <1+3i> [/itex]?

This means, is there some [itex] c+di [/itex] such that;

[itex] (c+di)(1+3i) = (a+(3a-10k))i [/itex]?

Yes, re-arranging shows that.

[itex] c = a - 3k, d = -k [/itex]

So [itex] (a+(3a-10k)) \in <1+3i> [/itex]

So [itex] ker(\theta) \subseteq <1+3i> [/itex]

This proves it for me, but the way my given answers simply say [itex]3a \equiv bmod10 \Rightarrow a+bi \in <1+3i> [/itex] suggests there is a clear way of noticing this without doing any working out. What is it?

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# Homework Help: Kernel of this homomorphism

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