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4 votes

De-diagonalize a diagonal matrix back over a subfield from the extension field

This is possible iff the characteristic polynomial of $D$ has coefficients in $K$, or equivalently iff the diagonal entries of $D$ are invariant under the action of the Galois group $G = \text{Gal}(L/...
Qiaochu Yuan's user avatar
2 votes

AES S-box as simple algebraic transformation

The notation $$s = \left(b \times 31_{10} \bmod{257_{10}}\right) \oplus 99_{10}$$ does correspond to a polynomial multiplication of $b$ times $f = 1+x+x^2+x^3+x^4 \in \mathbb F_{2}[x].$ Here $f$ is ...
Keplerto's user avatar
  • 503
2 votes

Finding polynomial with root $\sqrt{2}+\sqrt{3}$ over $\mathbb{Q}$, what is the degree of a root?

Since several posts are considered duplicates of this answer, I found it fitting to provide an answer giving a neat way to prove that the polynomial found in the previous answer is irreducible. The ...
SomeCallMeTim's user avatar
1 vote

What products (other than the complex product) can you define on $\mathbb{R}^2$ to make it a field? How about $\mathbb{R}^n$

It looks like Sammy Black's link addresses this, but I will still elaborate a bit: I suspect you may be interested in imposing operations which preserve the original structure of $\mathbb{R}^2$ ...
McNugget's user avatar

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