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5 votes
Accepted

Schur’s lemma over $\mathbb{F}_p$

Since we’re over a finite field, $\mathbb{End}_{\mathbb{F}_p[G]}(V)$ is also a field, but I am not sure that it is exactly $\mathbb{F}_p$, why couldn’t it be a field $\mathbb{F}_{p^n}$? In general, ...
Qiaochu Yuan's user avatar
3 votes

Linear algebra question: does it have a solution?

The problem is related to but not an exact match with linear complexity. I try and reuse some of the algebraic machinery from that theory as I recall it. My conclusion. The example sequence suggested ...
Jyrki Lahtonen's user avatar
3 votes
Accepted

Equivalent polynomials over a finite field

When looking at polynomials over $\Bbb F_p = \Bbb Z/p\Bbb Z$, there is a very nice explicit description of this equivalence between polynomials: Two polynomials $f(x),g(x) \in \Bbb F_p[x]$ take the ...
Greg Martin's user avatar
  • 81.7k
3 votes

Splitting Field of $x^4 - 10$ in $\mathbb{F}_7$

$$\begin{align}x^4-3 &= x^4+4 \\ &=(x^4+4x^2+4)-4x^2 \\ &= (x^2+2)^2-4x^2 \\ &=(x^2-2x+2)(x^2+2x+2)\end{align}$$ Note that $x^2\pm 2 x + 2 = (x\pm 1)^2+1$ has no root in $\mathbb F_7$, ...
Just a user's user avatar
  • 17.8k
2 votes

Is there a finite dimensional vector space over a finite field with exactly two bases?

To give this question an answer: Any $\mathbb{F}_3$-vector space $V$ of dimension $1$ has exactly two bases, namely the two singletons $\{x\}$ and $\{y\}$, where $x$ and $y$ are the two non-zero ...
azimut's user avatar
  • 23.1k
1 vote

Finding BCH code syndromes

The question matches the changes I made to my test code: $GF(2^4):1 + x^3 + x^4$, $m_1 m_3 = 1 + x + x^2 + x^4 + x^8$, $v(x) = x^2 + x^5 + x^8 + x^{11} + x^{14}$ I'm not aware of calculating syndromes ...
rcgldr's user avatar
  • 586
1 vote
Accepted

How many roots are there of $(x^2-3)(x^3-3)$ in $K$, where $K$ is the splitting field of $x^3-1$ over $\mathbb F_{11}$?

Hints: $x^n-1$ has $n$ distinct solutions in a field $\mathbf F$ iff the group of units $\mathbf F^*$ has a cyclic group of order $n$. Let $\mathbf F$ be a splitting field of $p(x)=x^n-k$ then $k$ is ...
Nothing special's user avatar
1 vote

Extended euclidian algorithm

The matrix form used in the wiki article is mostly a notation difference. The actual math is essentially the same. In order for the algorithm to function, at some point it needs to use $GF(2^4)$ ...
rcgldr's user avatar
  • 586
1 vote

Irreducible factors of $X^n -1$ in $\mathbb{F}_q[X]$

If $f$ is an irreducible factor of $t^n-1$ over $\mathsf k=\mathbb F_q$, then if $K$ is the splitting field of $g(t)=t^n-1$, $f$ splits in $K$, and hence if we take the subfield generated by one of ...
krm2233's user avatar
  • 5,023
1 vote

Irreducible factors of $X^n -1$ in $\mathbb{F}_q[X]$

I came up with a proof that uses less terminology but that may be different from what you brought up in your post [my eyes still glaze over at the words 'congruence class', I am sorry!] Proof #1: Let $...
Mike's user avatar
  • 21.1k
1 vote

Is every algebraic extension of a finite field Galois?

Yes, that is correct. Every algebraic extension of a finite field is Galois. Exactly because every finite extension of a finite field is cyclic and Galois. You seem to be knowledgable about the Galois ...
1 vote

Cohomology $H(X^{(p)}, \mathbb Q_{\ell})$ of the Frobenius twist of a variety over a finite field

If those are etale (small) cohomology groups, then they should be isomorphic by the topological invariance of the etale site (since the relative Frobenius morphism is a universal homeomorphism). See ...
Cristian D. Gonzalez-Aviles's user avatar

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