# Naïve groups, fields and ideals

Please excuse the simplicity of this question, but I am very new to groups and fields. I only seek an simplistic / intuitive expalnation, and confirmation / refutation re whether I am on the right track. I confess I am a little averse to reading about them from the ground up, as my motivation is quickly deteriorated when goin gthourough rigourous set theoretic foundations.I could put this down to different learning styles at best, laziness at worst! But I feel I learn best from concrete examples and pictoral representation - I am not at the abstract level of understanding yet really.

Rading Derbeshire's "Unknown quantity" he gives an example of and ideal in $\mathbb{N}$ as $\dots -60,-45,-30,-15,0,15,30,45,60\dots$

Does this extend to a quadratic ideal eg $\dots -32,-16,-4,-8,-2,0,2,4,8,16,32 \dots?$

Is it really this simnple, or am I missing something crcial, or have I got completely the wrong end of the stick? I am sure that there is much more to it than this, but is this a sound starting point to beginning to understand the modular nature of these things?

• Are you familiar with modular arithmetic or congruences?  e.g. $\,5+6\equiv 1\pmod{10}\quad$ – Gone Jun 18 '15 at 21:38
• @BillDubuque if you have any pointers in terms of modular arithmetic that might help my understanding at this early stage of groups, fiels & ideals, I would be most grateful! :) – martin Jun 19 '15 at 12:13

While that's actually a pretty clever idea, it's not what ideal means. First of all, Ideals need to be subrings, so if $16$ and $-4$ are in the ideal, then so must $16+(-4)=12$. Secondly, Ideals are meant to be "closed under multiplication." The idea is to generalize the notion of "has a factor of n." So, for any number $n$, we know that $n$ times something in (⋯−60,−45,−30,−15,0,15,30,45,60…) will have a factor of 15. The canonical example is even numbers. Anything times an even is an even. So the ideal $(...,-6,-4,-2,0,2,4,6,...)$ is the example you want to keep coming back to. The "quadratic ideal" your suggesting doesn't have this nice property, sadly.

This is a great question, by the way.

• thanks for the reply and the words of encouragement! You are right that I have to get to grips with commutivity and terms like "closed under multiplication". I will continue to persevere in thisd area ans I find it fascinating, but rather alien at the moment! – martin Jun 18 '15 at 21:35
• Rings are hard, and there are a lot of definitions. But it's really fun stuff. – Zach Stone Jun 18 '15 at 21:37
• @ZachSone It certainly looks like fun stuff, especially entering Galois representations, which are mostly unaccessible to me at the moment, yet it seems that this apporach yileds many beautiful results. – martin Jun 18 '15 at 21:59

You have an interesting thought. However, if you know about Normal Subgroups, you know that the kernel of a homomorphism is worth naming and study. An Ideal is useful because it is the Kernel of a homomorphism from one Ring to another.

This means an Ideal has to relate to both the additive and multiplicative structures. You have tried to capture the multiplicative idea, but, for example $2+4=6$ is not in your set, so it doesn't work for addition.

The work you put in at this stage in understanding will definitely pay off, so do keep asking questions until you have got it clear.

• thanks for your answer - yes, not you point it out addition is clearly not commutative in the quadratic example. I must admit modular mathematics is rather alien to me, thoiugh I understand it is the foundation of abstract algebra & many iportant results - I will certainly take your advice & persever, asking questions when necessary. It is comforting to know there is such a supportive community here to develop personal understanding :) – martin Jun 18 '15 at 21:32