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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
24
revised Finding the fixed Field of $\sigma \in Aut(\mathbb{R}(t)/\mathbb{R})$
corrected a typo
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
answered Is the degree of an infinite algebraic extensions always countable?
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
answered Hensel's lemma & $p$-adic polynomial roots
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.
May
22
comment How to find minimal polynomial of primitive element (field theory)
Precisely what do you mean by “primitive element”? Do you mean a generator of the multiplicative group, which is the interpretation of @GerryMyerson, or do you mean an element $\alpha$ such that $k=\Bbb F_2(\alpha)$?
May
21
comment Constructing a Fractional Linear Map
Yeah, I think that this definitely is not the right idea. Any such $\varphi$ will be a rotation of $120^\circ$ about some point.
May
20
answered Showing polynomial is irreducible over field containing roots of unity.
May
20
revised Constructing a Fractional Linear Map
added 118 characters in body
May
20
answered Constructing a Fractional Linear Map
May
19
comment Field extension over $\mathbb{Q}$
Um, @QiaochuYuan, I agree with you, once it‘s known that $E$ is finite over $\Bbb Q$. Don’t we have to mumble some magic words to argue that in fact the extension must be finite?
May
17
comment Solution of $8x^2+x+1=0\in \mathbb{Q}_2$.
@GerryMyerson, the good Hensel absolutely applies when the leading coeff is in the maximal ideal.
May
17
comment Show that $K = \mathbb{Q}(\sqrt{p} \ | \ \text{p is prime} \}$ is an algebraic and infinite extension of Q
But yet, @QiaochuYuan, his set is $\Bbb Q$-linearly independent, which is enough to show that the field is infinite over the rationals. On the other hand, the linear independence would need to be shown.
May
16
comment Working in $\mathbb Q[x]$. Two polynomials are coprime if their gcd is a constant?
This is very clear and complete.
May
16
comment Question on separable field extenions
You need a theorem that says that a compositum of separable extensions will itself be separable. Depending on your definition of separability, it can be easy to prove.
May
16
comment Explicit description of $\Bbb Q_p \cap \bar{\Bbb Q}$
Good. There’s also a theorem in Class Field Theory that says that if $K\supset k$ is any finite extension of number fields (in particular for $k=\Bbb Q$), there are infinitely many primes of $k$ that split completely in $K$. In other words, no matter the number field $K$, there are infinitely many $p$ such that $\Bbb Q_p$ contains a copy of $K$.