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I am a complex analyst who was been screwed over by fate and now has to work with elliptic curves for my doctoral dissertation. This entails learning about (non-category-theoretic) algebraic geometry. I'm having a terrible time, simply because whenever I sit down to do anything, I find myself utterly at a loss as to what to do. There are only definitions; there are no algorithms, or computations strategies for me to learn, and I'm losing my mind. I learn things by familiarizing myself with the patterns and rules of symbol manipulation in computations and proofs, until I've gotten enough experience with them to be able to understand things like definitions or arguments. Unfortunately, since every source I turn to assumes I can figure out the methods for manipulating symbols and performing computations, they have few worked out examples (usually none at all), and, even when they do, they skip over so many steps and justifications that I find myself even more confused than when I started.

That being the case, I want to tackle my difficulties one step at a time. Here, I wish to learn how to compute quotients of polynomial rings by ideals.

Let $mathbb{K}$
be a field. Given polynomials $f_{1},ldots,f_{N}$
, I write $leftlangle f_{1},ldots,f_{N}rightrangle$

to denote the ring/ideal generated by said polynomials over $mathbb{K}$. If I recall correctly (I know only very little of commutative algebra, and most of it I can't even remember properly). I suspect, but am not sure, that the following questions' answers might depend on the properties of $mathbb{K}$
(its characteristic, its algebraic closedness (or lack thereof), etc.) If so, how?

1) Let $r$
be a non-zero element of $mathbb{K}$ How do I compute $mathbb{K}left[x,yright]/leftlangle x^{2}+y^{2}-r^{2}rightrangle$
? (If I recall correctly, this object is a vector space, correct? As such, it should have a standard basis in terms of functions of $x$
and $y$. So, by “compute” I mean “what is the algorithm for obtaining the standard basis of the $mathbb{K}$
-vector space $mathbb{K}left[x,yright]/leftlangle x^{2}+y^{2}-r^{2}rightrangle$
?”)

2) How do I compute $mathbb{K}left[x,y,zright]/leftlangle x^{2}+y^{2}+z^{2}rightrangle$

3) How do I compute $mathbb{K}left[x,yright]/leftlangle x^{2},y^{2}rightrangle$

4) How do I compute $mathbb{K}left[x,yright]/leftlangle x^{3},yrightrangle$

5) How do I compute $mathbb{K}left[x,yright]/leftlangle x^{3}-y,x^{4}-y^{2}rightrangle$
?

6) How do I compute $mathbb{K}left[x,y,zright]/leftlangle x^{2}-z^{2},x^{2}-y^{2}rightrangle$

And so on.

Here is my attempt to do (1):

Using taylor series, any $fleft(x,yright)inmathbb{K}left[x,yright]$
can be written uniquely as $fleft(x,yright)=Pleft(xright)+yQleft(xright)+y^{2}Rleft(x,yright)$
for some $Pleft(xright),Qleft(xright)inmathbb{K}left[xright]$
and some $Rleft(x,yright)inmathbb{K}left[x,yright]$.

Since the quotienting out by $x^{2}+y^{2}=r^{2}$
tells me that $y^{2}=r^{2}-x^{2}$
in the resultant quotient space, I can write: $$fleft(x,yright)equiv Pleft(xright)+yQleft(xright)+left(r^{2}-x^{2}right)Rleft(x,yright)$$

Thus:

$$mathbb{K}left[x,yright] = left(mathbb{K}left[xright]right)oplusleft(ymathbb{K}left[xright]right)oplusleft(y^{2}left(mathbb{K}left[xright]right)left[y^{2}right]right)
equiv left(mathbb{K}left[xright]right)oplusleft(ymathbb{K}left[xright]right)oplusleft(left(r^{2}-x^{2}right)left(mathbb{K}left[xright]right)left[r^{2}-x^{2}right]right)$$

However, I do not know where to go from here. Worse still, I am unsure if this is even right, because the useless definition of the direct sum (the unique decomposition one) does not tell me what I want/need to know, which is/are the list of valid manipulations of the direct sum symbol, and expression that contain it. Same goes for the brackets in a notation like $mathbb{K}left[x,yright]$.

Any assistance with these difficulties would be much appreciated. Thanks in advance.

You could read about Groebner bases, but I don't think you need to know any theory to understand how to find basis of your algebras 1-6. For some you can even use elimination of variables. Generally speaking, this refers to any time you have a situation like $$K[x_1,ldots, x_n, y]/(f_1,ldots, f_r, y-f) cong K[x_1,ldots,x_n]/(f_1,ldots, f_r)$$ where $f$ is a function only of $x_1,ldots, x_n$ and not $y$. You can just eliminate the variable $y$ as it isn't contributing any new elements to the ring that the $x_i$'s didn't already express. So the ring is isomorphic to $K[x_1,ldots,x_n]/(f_1,ldots, f_r)$. (Note that if the $f_i$ had $y$ variable present, you have to plug in $f(x_1,ldots,x_n)$ in its place.)

(1) As a vector space it is $K[x] oplus yK[x]$. Anytime the power of $y$ goes above 1 you agree to write it in terms of $x$ instead. This means the basis is ${1,x,x^2,x^3,ldots, y, yx,yx^2,yx^3,ldots}$.

(2) As a vector space it is $K[x,y] oplus zK[x,y]$. Anytime the power of $z$ goes above 1 you rewrite it like $z^5 = z(x^2 + y^2)^2$ in terms of $x$ and $y$ instead.

(3) This is a finite dimensional algebra, with basis ${1, x, y, xy}$.

(4) we have a (very trivial) case elimination of variables where $f=0$. This ring is just $K[x]/(x^3)$ which is finite dimensional with basis ${1,x,x^2}$.

(5) You can eliminate $y$. This is instead $K[x]/(x^4 - (x^3)^2) = K[x]/(x^6 - x^4)$. This one is also finite dimensional, with basis ${1,x,x^2,x^3,x^4,x^5}$. Anytime the power of $x$ goes above $5$ you can subtract 2 from its exponent without changing the element of the ring. For instance $x^7 = x^5$ and $x^{10} = x^8 = x^6 = x^4$.

Note: In general $K[x]/(f(x))$ is an algebra of dimension equal to the degree $d$ of $f$ and ${1,ldots, x^{d-1}}$ is a basis.

(6) You can write $$K[x,y,z]/(x^2-z^2, x^2-y^2) = K[x] oplus yK[x] oplus zK[x] oplus yzK[x].$$ Anytime the power of $y$ or $z$ goes above 1, we can just change it to a power of $x$.