Skip to main content
MathOverflow

Return to Question

edited tags
Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286
added 19 characters in body
Source Link
Mikhail Goltvanitsa
Mikhail Goltvanitsa
  • 732
  • 3
  • 16

Hellow. I'm sure that the following is truth, but I can't prove it.

Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and $A = \{\theta\in K: \mathrm{ord}\theta = q^{mn}-1\}$ be the set of primitive elements. I want to prove the following statement. If $_RW$ be a subspace of $_RK$, $\mathrm{dim}_RW = n$ such that for every $\theta \in A$ $$ W\oplus W\theta\oplus\ldots\oplus W\theta^{m-1} = K, $$ then $W = S$$W = aS$ for some $a\in K$.

I will be grateful for the help.

Hellow. I'm sure that the following is truth, but I can't prove it.

Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and $A = \{\theta\in K: \mathrm{ord}\theta = q^{mn}-1\}$ be the set of primitive elements. I want to prove the following statement. If $_RW$ be a subspace of $_RK$, $\mathrm{dim}_RW = n$ such that for every $\theta \in A$ $$ W\oplus W\theta\oplus\ldots\oplus W\theta^{m-1} = K, $$ then $W = S$.

I will be grateful for the help.

Hellow. I'm sure that the following is truth, but I can't prove it.

Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and $A = \{\theta\in K: \mathrm{ord}\theta = q^{mn}-1\}$ be the set of primitive elements. I want to prove the following statement. If $_RW$ be a subspace of $_RK$, $\mathrm{dim}_RW = n$ such that for every $\theta \in A$ $$ W\oplus W\theta\oplus\ldots\oplus W\theta^{m-1} = K, $$ then $W = aS$ for some $a\in K$.

I will be grateful for the help.

Source Link
Mikhail Goltvanitsa
Mikhail Goltvanitsa
  • 732
  • 3
  • 16

Splitting subspaces and finite fields

Hellow. I'm sure that the following is truth, but I can't prove it.

Let $R<S<K, R=\mathrm{GF}(q),\ S= \mathrm{GF}(q^n), \ K= \mathrm{GF}(q^{mn})$ be a chain of finite fields and $A = \{\theta\in K: \mathrm{ord}\theta = q^{mn}-1\}$ be the set of primitive elements. I want to prove the following statement. If $_RW$ be a subspace of $_RK$, $\mathrm{dim}_RW = n$ such that for every $\theta \in A$ $$ W\oplus W\theta\oplus\ldots\oplus W\theta^{m-1} = K, $$ then $W = S$.

I will be grateful for the help.