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Apr
6
comment If $F$ has characteristic $p$ and $f(x)=x^p-a\in F[x]$, then $f(x)$ is either irreducible over $F$ or $f(x)$ splits in $F$.
The Frobenius homomorphism is used in (or rather mentioned to justify) the step $X^p-α^p = (X-α)^p$. Divisors of polynomials don’t change by expanding the field of coefficients – if $g ∈ F[X]$ is monic and $g | f$ in $F[X]$, then trivially $g | f$ in $L[X]$, hence $g = (X-α)^n$ in $L[X]$ for some $n ∈ ℕ_0$. Then you can expand $g$ in $L$ and conclude like I’ve done it.
Apr
6
comment Why $dz\wedge d\bar{z} = d|z|\wedge d\phi$ for $z \in \mathbb{C}$?
But what is the exterior derivative of a complex number?
Apr
6
comment Why $dz\wedge d\bar{z} = d|z|\wedge d\phi$ for $z \in \mathbb{C}$?
What does “$dz$” mean if $z$ is an actual complex number?
Apr
6
revised the number of inversions in the permutation “reverse”
added 6 characters in body
Apr
6
revised the number of inversions in the permutation “reverse”
added 6 characters in body
Apr
6
answered the number of inversions in the permutation “reverse”
Apr
6
comment Intuition for theorem about compact subsets of topological groups
So you do know a proof of the statement, but you still want some intuition for it, right?
Apr
6
revised If $F$ has characteristic $p$ and $f(x)=x^p-a\in F[x]$, then $f(x)$ is either irreducible over $F$ or $f(x)$ splits in $F$.
added 1 character in body
Apr
5
comment Career Advice: I love Abstract Algebra and Analysis… What should I do?
How do you feel about complex analysis?
Apr
5
comment If $F$ has characteristic $p$ and $f(x)=x^p-a\in F[x]$, then $f(x)$ is either irreducible over $F$ or $f(x)$ splits in $F$.
@OLP Yeah, I’ve updated it.
Apr
5
revised If $F$ has characteristic $p$ and $f(x)=x^p-a\in F[x]$, then $f(x)$ is either irreducible over $F$ or $f(x)$ splits in $F$.
expanded
Apr
5
comment Find the dimension of $\mathbb F[T]$ over $\mathbb F$
@quid »$ℚ[\sqrt{-5}]$« is short for »$ℚ[X]/(X^2 + 5)$«. On the other hand, »$F[T]$ with $m(X)$ being the minimal polynomial of $T$« isn’t short for »$F[X]/(m)$«.
Apr
4
comment I need a good reference in topology
Haven’t read much of it, but Bredon’s Topology is said to be a classic.
Apr
4
answered Let $\alpha\in R$. Prove that $\mathbb Q(\alpha)\cong\mathbb Q(x)$ iff $\alpha$ is transcendental.
Apr
4
comment Let $\alpha\in R$. Prove that $\mathbb Q(\alpha)\cong\mathbb Q(x)$ iff $\alpha$ is transcendental.
$ℚ[X] ≠ ℚ(X)$ if $ℚ[X]$ denotes the ring of polynomials @OLP. By the way, I recommend using capital letters if you are refering to indeterminates for polynomials/rational functions.
Apr
4
comment Prove that there is no irreducible polynomial in $\mathbb Q[x]$ which is zero at both $x=\sqrt 5$ and $x=\sqrt 7$
So you already know about Galois automorphisms and minimal polynomials and such?
Apr
4
comment If $F$ has characteristic $p$ and $f(x)=x^p-a\in F[x]$, then $f(x)$ is either irreducible over $F$ or $f(x)$ splits in $F$.
»Frobenius endomorphism« is just a fancy name for any map given by $x ↦ x^q$ whenever the Freshman’s Dream $(x+y)^q = x^q + y^q$ holds (so whenever the given map is additive/an endomorphism of an abelian group). Don’t worry about it, but keep it in mind because it’s very important.
Apr
4
comment If $F$ has characteristic $p$ and $f(x)=x^p-a\in F[x]$, then $f(x)$ is either irreducible over $F$ or $f(x)$ splits in $F$.
@OLP Don’t worry too much about the word »Frobenius homomorphism«, you already realized that the Freshman’s Dream holds and $f$ splits as $f = (X-α)^p$ (in its splitting field). I’ve added another hint. How do proper divisors of $f$ look like in $L$?
Apr
4
revised If $F$ has characteristic $p$ and $f(x)=x^p-a\in F[x]$, then $f(x)$ is either irreducible over $F$ or $f(x)$ splits in $F$.
added 41 characters in body
Apr
4
comment Prove that every open interval in $R$ is a union of at most countable collection of disjoint segments
And a segment would be …? A closed intervall possibly?