4
votes
2answers
377 views

Quotient of $Z[x_1,…,x_n]$ by a maximal ideal is a finite field [duplicate]

I am seeing the proof of the Ax-Groethendieck theorem from commutative algebra and I have a problem. How can I prove that if $x_1,...,x_n$ are complex numbers and $I$ is a maximal ideal of ...
3
votes
1answer
488 views

Order of an element in a finite field

Let $\mathbb F_p$ be the finite field of a prime order $p$, $f(x)\in \mathbb F_p[x]$ an irreducible polynomial, $E = \mathbb F_p[x]/\langle f(x)\rangle$ a finite extension of $\mathbb F_p$, ...
4
votes
3answers
568 views

Field with cyclic product group

If a field has a cyclic multiplicative group, is it necessarily finite?
4
votes
1answer
1k views

Irreducibility of some trinomials modulo $p$

Let $n>1$ be an integer. An old result of Selmer, See Theorem 1, page 289 in http://www.mscand.dk/article.php?id=1472, (If the link does not work try googling: ...
2
votes
2answers
307 views

vectors with entries from a finite ring

I've been working recently with vectors over finite fields, but I was hoping to work in a more general setting and consider vectors over finite commutative rings. The question I had is as follows: if ...
4
votes
1answer
490 views

Elements of trace zero in a field extension

Let $K=F_q$ and $F=F_{q^3}$, define the set A={$x \in F$ : $Tr_{F/K} (x)=0$}. Is it true that for every $x \in A$ there are $y,z \in A$ such that $x=yz$?
3
votes
1answer
794 views

Inverse for a permutation over GF(2)

Given a permutation $f: \{0,1\}^n \rightarrow \{0,1\}^n$ as $n$ polynomials over $GF(2)$ how to get formulas for the inverse permutation $f^{-1}$? I am interested in the answer to the previous ...