7 votes
Accepted

Complex numbers $z=a+ib$, what is the operation between $i$ and $b$

This is why formal definitions are so important. As you surely can find on this forum, a more rigorous definition is to define $\mathbb{C}$ as a set to be simply $\mathbb{R^2}$, and define two ...
Mark's user avatar
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4 votes
Accepted

Prove that $Char(F) \neq 0$ given that for all $x \in E$ there exists $n$ such that $x^n \in F$

I’m pretty sure that this argument isn’t optimal, but it gives a complete solution. I think you can also extend it to show more generally that in general, if $F$ is infinite, then $E/F$ is purely ...
Aphelli's user avatar
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2 votes

When working with $\mathbb Z_3[x] / \langle x^2+1\rangle$, why $(2x^2 + 2x + 1 + \langle x^2+1\rangle) = (2x + 2 + \langle x^2+1\rangle)$?

You're working modulo two things at once: mod 3 and mod $x^2+1$. Replacing a $2$ with a $-1$ which is allowed because we're working mod 3, we have $$2x^2+2x+1=-x^2+2x+1.$$ Adding $x^2+1$ which is ...
Gareth McCaughan's user avatar
2 votes

Complex numbers $z=a+ib$, what is the operation between $i$ and $b$

The precise meaning of $ib$ depends on the construction of the complex numbers. If they are defined as the set of numbers of the form $a+bi$ with multiplication defined as $(a+bi)(c+di)=ac-bd+bci+adi$ ...
Numeral's user avatar
  • 1,244
2 votes

Show $1+\sqrt{2}+\sqrt{3}+\sqrt{6}$ is a unit of $\mathbb{Q}(\sqrt{2},\sqrt{3})$.

I got $$ ( 1 + \sqrt 6)^2 - ( \sqrt 2 + \sqrt 3)^2 = 2 $$ so that
Will Jagy's user avatar
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1 vote
Accepted

Decomposition of $z^q-z$ with $q=p^n$ in $\mathbb{K}[x]$ with $|\mathbb{K}|=q$.

Hints. If $x\in K$ is not zero, what can you say about $x^{q-1}$ (Subhint: Lagrange) ? Deduce that every element of $K$ is a root of $P=z^q-z$ Conclude.
GreginGre's user avatar
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1 vote
Accepted

Reference for: Every element of a finite field is the sum of two squares.

Proposition 1 in Vitaly Bergelson, Andrew Best, and Alex Iosevich, Sums of Powers in Large Finite Fields: A Mix of Methods, The American Mathematical Monthly, 128:8, pp. 701-718. This has the ...
darij grinberg's user avatar
1 vote

Reference for: Every element of a finite field is the sum of two squares.

L'Arithmétique des corps, by Paulo Ribenboim. Page 182, chapitre X:"Sommes des carrés" Hermann París. Collection Méthodes. 1972. There is an English translation.
Piquito's user avatar
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