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9h
comment Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$
nice explanation of why $t$ certainly must be transcendental for the statement to be true.
1d
comment how to understand if the number is already squared or it should be squared in future?
Whether it’s reasonable or not, it reveals a lack of understanding that a good clean answer might remedy.
1d
comment In complex variables, why is |z-1| < 5 an open disk centered at +1, where the boundary is a circle of radius 5?
But you always get the distance between two things by looking at their difference.
1d
comment Inertia group modulo $Q^2$
What’s happening, as you can read in your text, is that, locally at $Q$, you have a map from the inertia group into $(\mathscr O_L/Q)^*$, by $\sigma\mapsto(\sigma\pi)/\pi$. A homomorphism, not necessarily one-to-one nor onto.
2d
comment Show that $\mathbb{Q}(\sqrt{2 +\sqrt{2}})$ is a cyclic quartic field i.e. is a galois extension of degree 4 with cyclic galois group
It’s clear that $\sigma\circ\sigma$ is identity; can you show that $\tau\circ\tau$ is $\sigma$?
2d
comment Inertia group modulo $Q^2$
Principalness of $Q$ is not the issue. See my response. Good question, by the way!
2d
comment Radical extension with root of cubic polynomial
Precisely what is a radical extension? Isn’t the field of seventh roots of unity a radical extension? It has a subfield cubic over $\Bbb Q$ that definitely is not gotten by adjoining the third root of something. If you’d like me to expand this in an answer, just give the word.
May
26
comment Why lower case “a” for “abelian group” and upper case “C” for “Cauchy sequence”?
Remember that in English we have a category that doesn’t exist in French: the proper adjective. The question then is only one about English; there’s no doubt that a pedant would object to “abelian group”. My upbringing was that the groups take his name uncapitalized, but Abelian varieties take the capital. Go figure.
May
26
comment Roadmap to $p$-adic numbers: where a self-learner should look for references
You might look into “Local Fields and their Extensions” by Fesenko and Vostokov.
May
26
comment Fields of characteristic $p$ that are isomorphic as vector spaces, but not as fields
Whoops: your examples are OK: the first is a genus-zero ff with constant field $\Bbb F_2$, the second is also genus zero, but the constant field is $\Bbb F_4$.
May
26
comment Fields of characteristic $p$ that are isomorphic as vector spaces, but not as fields
Zev, I think your two examples are both function fields of genus zero over $\Bbb F_2$. My examples would be $\Bbb F_2(T)$ itself and $\Bbb F_2(T)[Y]/(Y^2+Y+T^3)$. The latter is a ff of genus one, so definitely not isomorphic to the former.
May
26
comment Explain why $e^{i\pi} = -1$ to an $8^{th}$ grader?
This is my preferred argument. However, I once attempted to charm a second-term Calculus class in a liberal arts college with it, and the response was to ask me whether it was good for anything. If I had had my wits about me and the forethought, I would have prepared myself by being able to quote the Millay sonnet, “Euclid alone has looked on Beauty bare.”
May
25
comment Galois group of $ \mathbb{Q}(\varepsilon_5) / \mathbb{Q} $
The way to show that $(x^p-1)/(x-1)=\varphi(x)$ is irreducible is to expand $\varphi(x+1)$. Best to do it before accomplishing the division! (It Eisensteins.)
May
25
comment Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$
Oh sorry, blame an old man’s poor vision. I thought those were $1$’s up top, not $t$’s to the power $1$. Then to @Spooky I would precisely echo anon’s comment. (Again, sorry!)
May
24
comment Can a field extension still have “non-separability” above its maximal purely inseparable subextension?
Isn’t $K=F$ in this example?
May
24
comment Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$
@Spooky, that rational function is not the quotient of two odds, nor of two evens, so is not in $\Bbb R(t^2)$.
May
24
comment Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$
I just made the correction for @Spooky
May
23
comment Subgroups of automorphisms of Finite fields
You seem to be counting the automorphisms of your big field as finite group. But you need to look at field automorphisms, which are far fewer in number.
May
23
comment Hensel's lemma & $p$-adic polynomial roots
For $p=5$, Eisenstein applies, since all coeffs are divisible by $5$ and the constant is not divisible by $25$. So irreducible, no roots in $\Bbb Z_5$.
May
23
comment How to find minimal polynomial of primitive element (field theory)
The way to show that $X^6+X^3+1$ is irreducible over $\Bbb F_2$ is to observe first that any root of this polynomial is a primitive ninth root of unity; then to observe that you need to go all the way to $\Bbb F_{64}$ before you find any elements of multiplicative period $9$. Since the big field is of degree six over the little, your polynomial is irreducible.