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I am currently an part III student at Cambridge University, England.

I aim to answer what questions I can, understand some of those that I can't and improve the site however I can!


Dec
10
awarded  Excavator
Dec
8
awarded  Caucus
Dec
4
comment Irreducible components of affine variety
@Sodan Yes, that's what I meant.
Dec
3
answered Irreducible components of affine variety
Nov
18
comment Length of a module over a local Artinian ring
Which is exactly why my question is: does assuming $R$ is local make it easier to prove that any composition series has the same length? I appreciate that Jordan-Holder isn't "hard" but it seems harder than a throwaway exercise mentioned in lectures, so I assumed that the local condition made the problem easier (as it does in a lot of theorems).
Nov
18
comment Length of a module over a local Artinian ring
Was given in a lecture course. $M$ being finitely generated and $R$ Artinian means that we can use a result already proved as part of something else, that says if $M$ is a finitely generated Noetherian module, you can find a chain of submodules from $M$ to $0$ with each quotient isomorphic to $R/P$ for $P$ some prime ideal. If $R$ is Artinian, it is Noetherian so we can apply this, and then each $P$ is maximal, giving us a composition series. (For our purposes we did want such a series to exist). This explains all the assumptions in the set up above, other than why $R$ should be local...
Nov
18
comment Length of a module over a local Artinian ring
Thank you for the answer, but I am afraid that perhaps my question is unclear. What I would like to know is: given the setup above, is there an easy way to prove that any such composition series has the same length (using that $R$ is local) without resorting to the general case? Note that I did need to say the quotients were isomorphic to $R/P$ since I never stated they were simple (In this case the two properties are equivalent so I stated the more useful form). I am also aware of the general case, and how it simplifies to the local setup given above, but the point is that the setup above...
Nov
16
comment How to find the splitting field of $X^4-10X^2+1$?
The splitting field of a polynomial over a field (in this case I assume $\mathbb{Q}$) is the smallest field extension in which it splits. If $f(X) \in \mathbb{Q}[X]$ has roots $\alpha_1,\dots,\alpha_n$ then the splitting field is simply $\mathbb{Q}(\alpha_1,\dots,\alpha_n)$. Do you have any other requirements for what the splitting field should look like?
Nov
16
comment Length of a module over a local Artinian ring
@MartinBrandenburg Would it be possible for you to explain what you mean by "as usual" i.e. explain how this works in the local case?
Nov
15
comment Length of a module over a local Artinian ring
@MartinBrandenburg Yes, $R$ is commutative and unital. I'm not sure what you mean by "as usual" though, the proof I know comes from Eisenbud's "Commutative Algebra" which doesn't reduce to the local case. In fact, what I am looking for is exactly the proof of the local case. It might also be worth mentioning that I'm not sure why the local case implies the general one.
Nov
15
asked Length of a module over a local Artinian ring
Nov
8
answered Localization of a prime ideal in $\mathbb{Z}/6\mathbb{Z}$
Oct
27
awarded  Pundit
Oct
23
awarded  Yearling
Oct
17
reviewed Reject Why do we stop at exponentiation stage in arithmetic of natural numbers?
Oct
13
accepted Hausdorff property of $\mathbb{RP}^n$ from unusual definition
Oct
12
comment Hausdorff property of $\mathbb{RP}^n$ from unusual definition
Thanks, this is almost exactly what I wanted, and it relies solely on the fact that $\mathbb{R}$ is a field and a metric space, so even generalises to $\mathbb{C}$!
Oct
12
revised Hausdorff property of $\mathbb{RP}^n$ from unusual definition
edited title
Oct
12
comment Hausdorff property of $\mathbb{RP}^n$ from unusual definition
@HagenvonEitzen Yes, I don't think it's true that locally homeomorphic to $\mathbb{R}^n$ implies Hausdorff, Dan's comment above provides a counter-example.
Oct
12
asked Hausdorff property of $\mathbb{RP}^n$ from unusual definition