# Tagged Questions

Use this tag for questions about fields and field theory in abstract algebra. A field is, roughly speaking, an algebraic structure in which addition, subtraction, multiplication, and division of elements are well-defined. Please use (galois-theory) instead for questions specifically about that ...

2answers
49 views

### What are the “units” and “non-trivial divisors of zero” in a ring?

I'm confused on what units and non-trivial divisors of zero are when it comes to rings. For example, say I have this finite ring: R=GF(2)[x] mod x^3 + 1 = 0. Now I know the elements are 0, 1, x, x + ...
1answer
27 views

### Question about the zeros of $x^2+1 \in \Bbb Z_7.$

So obviously $6$ isn't a quadratic residue $mod(7)$ thus there are no zeros in $\Bbb Z_7$. So what I did next is considered considered the field $\Bbb Z_7[x]/<x^2+1>$ obviously $x+<x^2+1>$ ...
3answers
33 views

### Question about subfield of order p

I have seen this fact:- The minimal subfield of a field F of characteristic p is the field of p-elements. To me when I hear a finite field my mind directly go to think about the field of ...
1answer
45 views

### Showing $F_{\frac{p^2+1}{2}}\equiv p-1 \pmod{p}$ when $p\equiv \pm 2 \pmod{5}$ and $p\equiv 3 \pmod{4}$

A while back I was messing around with representations of finite fields and found this problem above while doing so. I'll explain below how I came to this point but my question is: Question: How ...
0answers
17 views

1answer
35 views

### To prove that in $K[x]$ , where $K=\mathbb Z_{p}(t)$, $f(x)=x^{p} -t$ is irreducible

$K=\mathbb Z_{p}(t)$ is the field of all rational polynomials over $\mathbb Z_{p}$ . The polynomial $$f(x)=x^{p} -t$$ has to be irreducible over $K[x]$. So the polynomial is in ...
1answer
35 views

### Characteristic property of field $K$

Given that $K$ is a field, $\text{char}(K)=p$ (where $p$ is prime) we need to show that for any integer $n$ the equality $$(a+b)^{p^n} =a^{p^n}+b^{p^n}$$ We have the following ...
0answers
32 views

### How we can solve this problem about primitive element?

I'm confused to find the primitive elements that we have in the below field: Let there be a finite field $F_4$ and a polynomial $ax+by+c$ such that $a,b$ are in set $S$ of primitive elements and ...
1answer
23 views

1answer
15 views

### $g(x) = x^p-x-\alpha \in k[x]$. Let $y$ be a root of $g(x)$ Show that $k(y)$ is separable normal over $k$.

Let $char(k)=p$ and $g(x) = x^p-x-\alpha \in k[x]$. Let $y$ be a root of $g(x)$ Show that $k(y)$ is separable normal over $k$. I have done many problems of this kind but this polynomial seems tricky. ...
1answer
30 views

### Suppose $\gcd(\deg(f),\deg (g))=1$ Show that $g(x)$ is irreducible in $k(\alpha)[X]$

This is an assignment. There are two related (I think) problems. Please solve one of them and I will try to solve the other. Let $\alpha, \beta$ be algebraic over $k$ whose irreducible polynomials ...
4answers
245 views

### Is there a relationship between vector spaces and fields/rings/groups?

I understand from a comment under Vector Spaces and Groups that every vector space is a group, but not every group is a vector space. Specifically, I would like to know, can I make a statement like: ...
1answer
225 views

0answers
25 views

### Infinite uncountable field as union of affine variety

Let $F$ be infinite uncountable field and $V(I)={(a_1,...,a_n) \in F^n | g(a_1,...,a_n) = 0 \forall g \in I}$. How can I prove what \$F^n = \bigcup_{i=1}^\infty V(I_i), V(I_i) \subseteq V(I_{i+1}), i ...