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19h
comment Practical example of non-homogenous recurrence relation
Do you mean a nonlinear recurrence? Googling the phrase "nonlinear recurrence" will give you results worth looking at.
1d
comment A two-variable quadratic form over a field of characteristic 2 with no nontrivial roots
The only thing I used about $F$ being finite is that we are guaranteed an irreducible quadratic in $F[x]$ exists. (I'm not even using anything about the characteristic being 2 or not.) Even if $F$ is infinite, as long as $F$ has a quadratic irreducible the same construction works. There are certainly $F$ for which there are no irreducible quadratics, such as algebraically closed fields, which exist in every characteristic.
1d
comment A two-variable quadratic form over a field of characteristic 2 with no nontrivial roots
For a finite field of char. 2 let $x^2 +bx+c$ in $F[x]$ be irreducible. Then $x^2 + bxy + cy^2$ on $F^2$ vanishes only at $(0,0)$. For example, if $F = F_2$ then you can use $b=c=1$.
Jan
24
comment Why does vector sum $(x_1,x_2)+'(y_1,y_2)=(x_1+2y_1, 3x_2-y_2)$ and $(cx_1,cx_2)$ fail to hold the axiom of vector space?
You gave us a formula for it! So it "works" by using it: substitute in actual numbers for the coordinates on the left and the formula on the right tells you what it is. Do you not see how the function $f(a,b) = a+3b$ "works"?
Jan
24
comment Why does vector sum $(x_1,x_2)+'(y_1,y_2)=(x_1+2y_1, 3x_2-y_2)$ and $(cx_1,cx_2)$ fail to hold the axiom of vector space?
You should not mix ordinary vector addition with the vector operation +'. Check if $(\mathbf x +' \mathbf y) +' \mathbf z = \mathbf x +' (\mathbf y +' \mathbf z)$. That is not what you're doing.
Jan
24
comment Is “Generalized functions” by Gelfand published in 5 or 6 volumes?
@maojun1998, you now have the title, so how you choose to acquire a copy (for free or by purchasing it) is your business. The download links on the page I mentioned are pretty obvious, but if you don't read Russian then they're irrelevant to you.
Jan
24
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Jan
23
revised Is “Generalized functions” by Gelfand published in 5 or 6 volumes?
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Jan
23
answered Is “Generalized functions” by Gelfand published in 5 or 6 volumes?
Jan
23
answered Why is a linear transformation called linear?
Jan
5
awarded  Nice Answer
Dec
30
comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$
I agree with Tom that the squaring of a sum of 4 terms starts to lead into some nasty algebra, and since we shouldn't bring in ideas that are beyond the scope of a basic Galois theory course or are too computationally intensive for an exam (trace mappings, quartic discriminants, ramification in number fields, etc.) I suspect this is not the kind of solution that would be used on a prelim exam.
Dec
30
comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$
Concerning $\sqrt{11}$ not being in $\mathbf Q(\sqrt{5})$ also, my point was that what I said follows from the lack of a real embedding of $\sqrt{-11}$, not that it is equivalent to this. So your comment is not inconsistent with mine.
Dec
30
comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$
OK, that's a type of "real number trick" that you invariably wind up using with these exam-type problems where the full classification of Galois groups of quartics is unreasonable to bring to bear.
Dec
30
comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$
For the record this requires showing first that $\mathbf Q(\sqrt{5},\sqrt{-11})$ has degree 4 over $\mathbf Q$, which is equivalent to showing $\sqrt{-11} \not\in \mathbf Q(\sqrt{5})$, and that follows from $\mathbf Q(\sqrt{5})$ having a real embedding while $\sqrt{-11}$ of course can't be embedded into $\mathbf R$.
Dec
29
comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$
Since this is meant to be a "basic" question I am wondering if there might be a typographical error, e.g., perhaps the polynomial should be $x^4 - 3x^2 - 5$. I am suggesting this because in these types of exam questions about Galois groups of quartics in $\mathbf Q[x]$ you can usually do some trick about real vs. nonreal roots to make progress in place of using more machinery. With constant term $-5$ such a trick is available (the quartic has real and nonreal roots), but with constant term $5$ this trick doesn't work.
Dec
29
comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$
Where is this question coming from (old qualifying exam?), and what is the assumed background for solving this problem, e.g., are you not allowed to use algebraic number theory?
Dec
29
comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$
It is false that $x_1^2 = x_2^2$: otherwise $x_2 = \pm x_1$, but $x_2$ is neither $x_1$ nor $-x_1$.
Dec
29
comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$
Let $r$ be a root of $f(X)$. Show $[K:\mathbf Q(r)] = 2$ by finding a root $s$ of $f(X)$ that is not in $\mathbf Q(r)$ and checking that all the roots of $f(X)$ are in $\mathbf Q(r,s)$ and that $s$ is quadratic over $\mathbf Q(r)$.
Dec
29
revised Why does topology rarely come up outside of topology?
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