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 ...

learn more… | top users | synonyms (1)

2
votes
2answers
1k views

Prove that every nonzero prime ideal is maximal in $\mathbb{Z}[\sqrt{d}]$

$d \in \mathbb{Z}$ is a square-free integer ($d \ne 1$, and $d$ has no factors of the form $c^2$ except $c = \pm 1$), and let $R=\mathbb{Z}[\sqrt{d}]= \{ a+b\sqrt{d} \mid a,b \in \mathbb{Z} \}$. ...
0
votes
0answers
45 views

Why is it true that $F_{q^n} = F_q(\alpha)$ where $\alpha$ is the primitive element of $F_{q^n}$?

Since $\alpha$ is the primitive element of $F_{q^n}$ then $F_{q^n} = \{0, \alpha, \alpha^2,\cdots, \alpha^{q^{n -2}} , 1\}$. Then how $F_q(\alpha)$ is equivalent to $F_{q^n}$? Because what I ...
1
vote
1answer
126 views

Conditions under which a variety to remains smooth after base change (if p > 0)

Let $k$ be an arbitrary field of positive characteristic and let $V$ be a smooth projective (irreducible) variety over $k$. Suppose that $K/k$ is a field extension such that $V_K:=V\times_{\text{Spec ...
0
votes
2answers
47 views

Does there exist a ring containing $k \times k$ that is algebraically closed with respect to $k[x,y]$?

Let $k$ be a field. We know that there exists a field $\bar{k}$ that is an algebraic closure of $k$ with respect to the polynomial ring $k[x]$. But does there exist a ring containing $k^2$ in which ...
1
vote
1answer
77 views

What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$

I'm doing some exercises to prepare for my exam: What is the splitting field of $X^{20}-1$ over $\Bbb F_3$. And how to factor $X^{20}-1$ in $\Bbb F_3[X]$. I've no idea how to tackle this ...
4
votes
0answers
178 views

Ultraproduct of the algebraic closure of finite fields

Ok, I have done a little research on the next problem, and there are simple proofs of it using model theory, logic and the Łoś's theorem. But here, the idea is to prove it using only Field Theory, ...
3
votes
1answer
141 views

Linear transformation whose $n$th power is identity

Let $V$ be a vector space over field $F$ with $\dim_FV=2$. Suppose $T:V\longrightarrow V$ is a linear transformation with $T^n=Id$ for some positive integer $n$ (the finite $n$ is the order of $T$). ...
1
vote
1answer
111 views

$\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$

I want to prove: $\sqrt[7]{11}$ is not contained in the splitting field of $x^7-12$ over $\mathbb{Q}$. Is there any direct way to prove? I have computed that the splitting field of $x^7-12$ ...
0
votes
2answers
147 views

Intersection of splitting fields of two polynomials

Let $f,g\in \mathbb{Z}[x]$, $(f,g)=h\in \mathbb{Z}[x]$ (if this is not true, can we construct $h$ in another way?). Let $K_1,K_2$ be the splitting fields of $f,g$ over $\mathbb{Q}$ respectively. Is ...
0
votes
2answers
181 views

Perfect field of characteristic $p>0$ which is not an algebraic extension of the prime field

True/False If $K$ is a perfect field of characteristic $p>0$, then is $K$ algebraic over $\mathbb{F}_p$? My guess is no and I try to find a counterexample for this. Can anyone give me some ...
0
votes
2answers
441 views

Cardinality of algebraic extensions of an infinite field.

An exercise in Lang's algebra book is: let $k$ an infinite field, and $E$ an algebraic extension of $k$. Then $E$ has the same cardinality as $k$. How can one can prove this?
1
vote
3answers
93 views

polynomial over a finite field

Show that in a finite field $F$ there exists $p(x)\in F[X]$ s.t $p(f)\neq 0\;\;\forall f\in F$ Any ideas how to prove it?
1
vote
1answer
31 views

Simultaneous irreducibility of minimal polynomials

Let $F$ be a field. Let $u,v$ be elements in an algebraic extension of $F$ with minimal polynomials $f$ and $g$ respectively. Prove that $g$ is irreducible over $F(u)$ if and only if $f$ is ...
7
votes
1answer
151 views

Short method to prove the irreducibility of $x^7+21x^5+35x^2+34x-8$ over $\Bbb Q$

I am given a task to prove that polynomial $f=x^7+21x^5+35x^2+34x-8$ is irreducible over $\Bbb Q$. In my algebra course we learnt reduction and Eisenstein criterion. Eisenstein doesn't seem to work ...
1
vote
1answer
52 views

Newbie material on field theory

I'm studying non-linear systems on my own. I have a basic idea of field diagrams for linear systems in 2d, although I'm not fully grounded in this. Are there any tutorials or material that you would ...
2
votes
0answers
44 views

Intermediate field of $\Bbb Q(\alpha)$ and $\Bbb Q$ [duplicate]

Let $f$ be an irreducible polynomial of degree 4 over $\Bbb Q$ and $Gal(f)=S_4$. Prove that there isn't nontrivial intermediate field between $\Bbb Q(\alpha)$ and $\Bbb Q$ where $\alpha$ is a root of ...
1
vote
1answer
85 views

function of a unit in a Euclidean domain

Let $R$ be a Euclidean domain and let $u$ be a unit in $R$. If we denote $\delta$ the corresponding function, is it true that $\delta(c)=\delta(uc)$ for every non-zero $c \in R$? I know that an ...
1
vote
3answers
1k views

Galois group of an irreducible polynomial

Find the Galois group of the polynomial $x^5-9x+3$ over $\mathbb{Q}$. since $3$ cannot divide $a_5$, $3$ divide other coefficients, $3^2$ cannot divide $a_0$, we see that the polynomial is ...
3
votes
1answer
186 views

Find a finite extension of $\mathbb F_2(x,y)$ having infinitely many intermediate subfields.

Let $\mathbb F_2$ be a field of order $2$. Let $F$ be the function field $\mathbb F_2(x,y)$, i.e., $x,y$ are independent variables and $F$ is the field of quotients of the polynomial rings $\mathbb ...
1
vote
2answers
193 views

Is the following set (with the usual addition and multiplication of numbers) a field?

Is the follwing set a field? $Q ∪ [-1,1]$ I read the notes in my textbook about fields and understand most of the field axioms. However I want to see an example which is worked. Furthermore I am ...
2
votes
1answer
456 views

Find all solutions for a system of linear equations over a given field

My Problem is: to find all Solutions for the following given System of linear equations over the Field $K = \mathbb{Z}_{/7}$ The System is given with: $$\begin{equation} \begin{split} ...
0
votes
1answer
69 views

Solving a linear equation in an extension field

Let $\mathbb K$ a commutative field and $\mathbb L$ an extension field of $\mathbb K$ (that is to say a field that contains $\mathbb K$). Let $A$ be a $n \times p$ matrix over $\mathbb K$. Let $B$ ...
1
vote
2answers
71 views

Let $\alpha \in \overline{\Bbb Q}$ a root of $X^3+X+1\in\Bbb Q[X]$. Calculate the minimum polynomial of $\alpha^{-1}$ en $\alpha -1$.

Let $\alpha \in \overline{\Bbb Q}$ a root of $X^3+X+1\in\Bbb Q[X]$. Calculate the minimum polynomial of $\alpha^{-1}$ en $\alpha -1$. I don't really understand how to get started here. I know ...
2
votes
2answers
166 views

Let $f:K \to L$ a homomorphism of fields. Prove that $f$ induces a isomorphism of the prime fields of $K$ and $L$.

Let $f:K \to L$ a homomorphism of fields. Prove that $f$ induces a isomorphism of the prime fields of $K$ and $L$. Here are my thoughts: Assume $K$ is infinite. Then $L$ must be infinite as ...
4
votes
1answer
228 views

Solvable by radicals or not?

Let $f=2x^5-5x^4+5\in \Bbb Q[x]$. Then, how to prove that it's not solvable by radicals? Since $f$ is solvable by radicals iff $Gal(f)$ is solvable and $Gal(f)\subset S_5$ (up to isomorphism), I ...
4
votes
2answers
92 views

Well known problem about irreducibles

I think this is well known, but I can't find a proof anywhere. Any help is appreciated. If $f(x)$ is monic and irreducible in $\mathbb{F}_p[x]$ with degree $m$, then $f(x)\mid x^{p^m}-x$.
3
votes
1answer
136 views

Model theoretic answer for having algebraic closure

I am beginner at the model theory and I learn compactness theorem at the class and I saw some application of it and one of them is that "every field has an algebraic closure". How can I prove it with ...
3
votes
1answer
143 views

Vector space isomorphic fields

$\Bbb R(X)\simeq\Bbb R(X^2)$ but $[\Bbb R(X):\Bbb R(X^2)]\neq [\Bbb R(X):\Bbb R(X)]$. Is this correct? I thought the dimension of a vector space remains the same if I replace the field by an ...
3
votes
1answer
683 views

Find the degree and a basis for $\mathbb{Q}( \sqrt2, \sqrt3)$ over $ \mathbb{Q}( \sqrt2 +\sqrt 3)$

Similar questions were asked here before but I still can't find exactly what I want. I want to find the degree and a basis for $\mathbb{Q}( \sqrt2, \sqrt3)$ over $ \mathbb{Q}( \sqrt2 +\sqrt 3)$ I've ...
2
votes
2answers
176 views

Extend a rational number field $\mathbb{Q}$ by using a transcendental number?

Here denoting a set of real transcendental numbers $\mathbb{T}$, what can we then say about the structure $$ \mathbb{Q}(t) = \left\{\, \sum_{k=0}^{+ \infty} a_k t^k\mathrel{}\middle|\mathrel{} a_k \in ...
0
votes
1answer
25 views

a question on spliting field

Let $f$ be a polynomial of positive degree over a field $F$ and $E$ the spliting field of $f$ over $F$ , do there exist some elements $a_1, a_2, ,…,a_s $ of $F$ and positive integers $n_1,n_2, ...
1
vote
1answer
32 views

Let F⊆K be fields and let f and g be polynomials in F[x]. If f is irreducible in K[x], show that it is irreducible in F[x].

Let $F \subseteq K$ be fields and let $f$ and $g$ be polynomials in $F[x]$. If $f$ is irreducible in $K[x]$, show that it is irreducible in $F[x]$.
1
vote
0answers
31 views

Finding the smallest $k$ such that $f(x)$ divides $1-x^k$ where $f(x)$ is over $\mbox{GF}(2)$?

One technique is iterative that is to assume alpha as the root and solve for a higher exponent ($x$) until $\alpha^{x} = 1$. Is there any other technique?
1
vote
1answer
396 views

Finding a primitive element of a finite field

Let $F = \mathbb{F}_p[x]/(m(x))$, where $m(x)$ is irreducible in $\mathbb{F}_p[x]$. How do I find a primitive element of $F$, i.e., one that generates the nonzero elements of $F$ multiplicatively? ...
1
vote
0answers
67 views

Cyclotomic polynomials to find the subgroups of a Galois group

With $f(x) = x^{10}+1$, I want to draw the lattice of subgroups of the group $Gal(L/\mathbb{Q})$. Using cyclotomic polynomials I find that we have the $Gal(\mathbb{Q}(e^{\frac{2 \pi i}{20}}) / ...
1
vote
1answer
99 views

Galois groups of intermediate fields

Suppose $k\subset E$ is Galois and let $F$ and $F'$ be two intermediate fields. Let $FF'$ be the smallest intermediate field containing $F$ and $F'$. Also let $G$ denote $\text{Gal}(E/k)$. Let ...
1
vote
0answers
32 views

Is there any vanishing criterion for elements of a tensor product of algebras?

Let $k \subset L$ be a field extension, $\mathrm{char}(k) = p$. I have some polynomial $f(X_1, \ldots, X_n) \in k^{1/p}[X_i], f^p \in k[X_i]$, with at least one coefficient not in $k$, such that ...
0
votes
0answers
50 views

Finding the splitting field of a function that is not trivial

I have a splitting field question, but I will try my best at attempting the problem to the best of my ability. Consider the function $f(x) = x^{10} + 1$. I want to find a primitive element of the ...
13
votes
5answers
221 views

Conceptual reason why a quadratic field has $-1$ as a norm if and only if it is a subfield of a $\mathbb{Z}/4$ extension?

I have convinced myself in a computation-heavy, ad-hoc way that a quadratic extension $K$ of $\mathbb{Q}$ occurs as the unique quadratic subfield of a $\mathbb{Z}/4$-extension of $\mathbb{Q}$ if and ...
1
vote
1answer
99 views

Primitive elements of an extension L/K and the K-embedding's of L

I was studying for my preliminary exams and found this problem on a previous exam. I think I proved the forward implication, but I am having trouble proving the converse statement. Any help would be ...
5
votes
1answer
162 views

Prove that $\mathbb{Q}(\alpha) = \mathbb{Q}(\alpha^{2})$

Let $f = X^{4}+X^{3}-X+2 \in \mathbb{Q}[X]$ and suppose that $f(\alpha)=0$ with $\alpha \in \mathbb{C}$. Prove that $\mathbb{Q}(\alpha) = \mathbb{Q}(\alpha^{2})$. So far i've tried to look at ...
3
votes
1answer
75 views

Degree of sum of algebraic elements

Consider a field extension $F$ of $K$. Let $u, v \in F$ be algebraic elements over $K$ (so that there exists two nonzero polynomials $f(x), g(x) \in K[x]$ such that $f(u)=g(v)=0$). I am interested ...
1
vote
0answers
87 views

$F$ a finite field of $p^n$ elements. Suppose $F^\times=\langle x \rangle$. Then $\phi(x)=x^{p^r}$ for automorphisms

Let $p$ be a prime and $F$ a finite field of $p^n$ elements. Suppose $F^\times=\langle x \rangle$. Let $\phi$ be an automorhpism of $F$. Then prove that $\phi(x)=x^{p^r}$ for some integer $r$. How to ...
0
votes
1answer
62 views

field extension $F(x)=F(x^2)$ [duplicate]

Let $x$ be algebraic over $F$ such that the field extension $F(x):F$ satisfies $[F(x):F]$ odd. Then prove $[F(x):F]=[F(x^2):F]$ hence $F(x)=F(x^2)$. How to prove? I only obtained the proof for the ...
1
vote
1answer
127 views

finding the number of sub fields such that $(K : Q) = 2$

Consider the polynomial $f(x) = x^5 - 4x + 2$. Let $L$ be the complex splitting field of $f(x)$ over $\mathbb{Q}$. I want to find the number of subfields $K$ of $L$ such that $(K : \mathbb{Q}) = 2$. ...
1
vote
1answer
31 views

Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$.

Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$. It suffices to prove for the case $F=\mathbb{Z}_p$. How to prove?
0
votes
1answer
191 views

Prove that $\mathbb Q(\sqrt[3]2), \mathbb Q(w\sqrt[3]2), \mathbb Q(w^2\sqrt[3]2)$ where $w = (-1+i\sqrt3)/2$ are distinct fields

I know that $\mathbb Q(\sqrt[3]2), \mathbb Q(w\sqrt[3]2), \mathbb Q(w^2\sqrt[3]2)$ are isomorphic to each other. However, I don't quite see how we can show that in fact they are distinct. The book ...
0
votes
1answer
36 views

Fields and irreducible polynomial of $p^n$ degree

Let $K$ be a field of $p$ elements. Let $f(x) \in K [x]$ be an irreducible polynomial of degree $n$. Prove that the field $K[x]/(f(x))$ has $p^n$ elements. By given theorem, let $K$ be a field, ...
1
vote
2answers
411 views

minimal field extension of Q($\sqrt[3] {2}$)

I need to describe the minimal field extension $\mathbb Q(\sqrt[3] {2})$ of the rational numbers $\mathbb Q$ that contain $\sqrt[3] {2}$. $\mathbb Q(\sqrt[3] {2}) =\{a+b\sqrt[3] {2}+c(\sqrt[3] ...
3
votes
3answers
1k views

Prove that Q($\sqrt{2}$, $\sqrt{3}$) is a field

Prove that $\mathbb{Q}(\sqrt{2}, \sqrt{3}) = \{a+b\sqrt{2} +c\sqrt{3} +d\sqrt{6}\ |\ a,b,c,d \in \mathbb{Q}\}$ is a field. I am doing the subfield test, but having trouble in showing how to express ...