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 ...

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2
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1answer
31 views

Necessary condition for the extension $F(\alpha) /F$ to be algebraic?

Let $K/F$ be an extension of fields and let $\alpha$ be an element of $K$ such that the set $\quad X = \{ \varphi(\alpha) \; | \; \varphi \in \operatorname{Aut}(K/F) \}$ is finite. Then, if $K/F$ ...
0
votes
0answers
21 views

The fundamental theorem of Galois theory

Let $E/Q$ be a Galois extension of degree $p^2$, where $p$ is a prime number. Prove that $L/Q$ is a Galois extension for any $L \in Intermediate(E/Q)$ and find $p$ if the cardinality of ...
2
votes
1answer
44 views

Find irreducible polynomial over $\mathbb F_9$

I am looking for a polynomial of degree $3$ in $\mathbb{F}_{9}$. How do I find one ? And if I have one how do I show that it is irreducible ? I would start with an irreducible polynomial in ...
0
votes
1answer
23 views

on a proof of the Primitive Element Theorem in zero characteristic

Here is a version of the Primitive Element Theorem and a proof in Fulton's Algebraic Curves: Question: My interpretation of the notation $(H,F)=(T-x) \in K'[T]$ is that the greatest common divisor ...
-2
votes
1answer
25 views

Integral closure and field of fractions

I have a ring $R = \mathbb{Q}[t^2,t^5] \cong \frac{\mathbb{Q}[x,y]}{\langle x^5 - y^2 \rangle}$ (where the denominator is the ideal generated by $x^5 - y^2$). Now i have to compute the closure of $R$ ...
2
votes
2answers
49 views

Etingof Problem 5.1, “Field embeddings”

Recall that $k(y_1, \dots, y_m)$ denotes the field of rational functions of $y_1, \dots, y_m$ over a field $k$. Let $f : k[x_1, \dots, x_n] \to k(y_1, \dots, y_m)$ be an injective $k$-algebra ...
2
votes
1answer
44 views

Why is the algebraic closure of $\Bbb{Q}$ not a finitely generated $\Bbb{Q}$-module?

Let $\overline{\Bbb{Q}}$ be the algebraic closure of $\Bbb{Q}$. I am trying to show that $\overline{\Bbb{Q}}$ is not finitely generated as a $\Bbb{Q}$-module, however I do not know where to go with ...
1
vote
2answers
35 views

Can we find the generator of the Galois group of $x^{p-1}+x^{p-2}+…1$?

$p$ is a prime. We know that $x^{p-1}+x^{p-2}+...1$ is irreducible in Q[x]. And the splitting field of $x^{p-1}+x^{p-2}+...1$ over $Q[x]$ is $Q(\xi_p)$-the primitive pth root of unity. Now I want to ...
1
vote
1answer
32 views

Is there a field $K[\alpha]$ with $\alpha$ idempotent?

I have a field $K$ and an idempotent element $\alpha\not\in K$ (i.e., $\alpha^2=\alpha$), and I would like to form the ring $K[\alpha]$. Can this structure exist (1) in general and (2) with ...
1
vote
1answer
22 views

$F$ be a field of non-zero prime characteristic $p$ , is it true that there is only one group homomorphism $f:(F,+) \to (F$ \ $\{0\},.)$?

Suppose $F$ be a field of non-zero characteristic $p$ , is it true that there is only one group homomorphism $f:(F,+) \to (F$ \ $\{0\},.)$ ? I have tried taking $x \in F$ , then $px=0$ , so ...
2
votes
2answers
23 views

Showing that a given field is the splitting field of a given polynomial

Let $F = Z_2$; show that the splitting field of $f(x) = x^3 + x^2 + 1 \in F[x]$ is a finite field with $8$ elements. As $f$ has degree $3$, it is reducible if it has root in $F = Z_2$ but by ...
0
votes
1answer
17 views

Let $K\subset L$ a finite field extension and $G=G(L/K)$ a Galois group of $L$ over $K$ then $|G|\leq[L:K]$.

Let $K\subset L$ a finite field extension and $G=G(L/K)$ a Galois group of $L$ over $K$ then $|G|\leq[L:K]$. I was trying to show this. By the Primitive Element Theorem $\exists\alpha\in L$ such ...
0
votes
0answers
26 views

If P(X) is reducible in K[X], show it is reducible in A[X], A integrally closed domain

Let $A$ be an integrally closed ring, $K$ its field of fractions, and $P(X) \in A[X]$ a monic polynomial. If $P(X)$ is reducible in $K[X]$, show that it is reducible in $A[X]$. The hint given is to ...
0
votes
1answer
33 views

Proof by contradiction, fields

Given the field $\mathbb{K}:=\{a+b\sqrt{2}: a,b\in \mathbb{Q}\}$, how would I prove that every $x\in \mathbb{K}$ is uniquely representable in this way: $x=a+b\sqrt{2}$, with $a,b\in \mathbb{Q}$? I ...
3
votes
1answer
25 views

Is it true that all the roots of the minimal polynomial of $\alpha$ are Galois Conjugates of $\alpha$?

To be more precise, Suppose [K:F] is Galois, $\forall \alpha \in K$ $\alpha \not\in F$, let $m_\alpha$ be the minimal polynomial of $\alpha$ in $F[x]$. I know that $\phi \in Gal(K:F)$, $\phi(\alpha)$ ...
1
vote
1answer
72 views

Subring generated by $x$ is an integral domain iff it is a field iff the minimal polynomial of $x$ is irreducible

Let $R$ be a ring, $K$ a subfield of $R$, and $x \in R$. Let $F(X)$ be the minimal polynomial of $x$ over $K$. I want to prove that: $K[x]$ is a field $ \iff K[x]$ is an integral domain $\iff ...
4
votes
1answer
54 views

Lattice of intermediate fields of $\mathbb{Q}(i,\sqrt{2},\sqrt{3}) / \mathbb{Q}$

I need to construct the lattice of intermediate fields of $\mathbb{Q}(i,\sqrt{2},\sqrt{3}) / \mathbb{Q}$. I think I know how to do this by simply listing them, but it seems that the picture I get ...
2
votes
3answers
57 views

To Factorize $x^{27}-x$ over $\mathbb F_3$.

Problem 7.5 in Chapter 15 of Artin's Algebra asks to factorize $x^{27}-x$ over $\mathbb F_3$. Here is what I have done. $x^{27}-x=x(x^{26}-1)= x(x^{13}-1)(x^{13}+1)$. In am having trouble ...
3
votes
1answer
64 views

To prove: $ [K : \mathbb{Q}] = 2 \ \Longrightarrow \ \exists \zeta \text{ primitive root of unity}, \ \mathbb{Q}(\zeta) \ \supseteq \ K $

I have to show that the following statement is true: Let $K$ Be a field extending $\mathbb{Q}$ such that $[K: \mathbb{Q}] \ = \ 2$. Then there is a root of unity $\zeta$ such that $K \subseteq ...
6
votes
4answers
103 views

Show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$

I want to show that $\sqrt{2}\notin \mathbb{Q}(\sqrt[4]{3})$. I think it would be easier to prove it using the following: $\mathbb{Q}\subset\mathbb{Q}(\sqrt{3})\subset\mathbb{Q}(\sqrt[4]{3})$. Then ...
1
vote
3answers
38 views

Why does F(X) divide F'(X)

Let $K$ be a field of characteristic zero or a finite field, let $F(X) \in K[X]$ be a monic irreducible polynomial, and let $F(X)= \prod_{i=1}^n(X-x_i)$ be its decomposition in an extension $K'$ of ...
2
votes
0answers
30 views

Find the coefficients in $(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 = s_1^2s_2^2 + as_1^3s_3 + bs_1s_2s_3 + cs_2^3+ds_3^2$

Use evaluation homomorphisms $F[x_1,x_2, \dots, x_n] \to F$ to obtain the coefficients in: $$(x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2 = s_1^2s_2^2 + as_1^3s_3 + bs_1s_2s_3 + cs_2^3+ds_3^2$$ where the $s_i$ ...
0
votes
1answer
36 views

Proving the following set of real numbers is a field

Show that the following set $A$ of real numbers under addition and multipication is a field: $A = {a + b\sqrt{2} : a,b \ \text{rational}}$ I am not sure if I am right but here is what I have thus ...
0
votes
1answer
39 views

Writing elements of a finitely generated field extension in terms of the generators

I have a question that arose in the process of solving a different problem in Galois theory. That problem asked to show that if $\alpha_1, \dots, \alpha_n$ are the generators of an extension $K/F$, ...
0
votes
1answer
32 views

Any case other than $0$ for $a=-a$

I was wondering about a rather simple type of problem. This is not any sort of homework but I was working on problems in linear algebra and in any cases that we have $a=-a$ we can simplify to then ...
2
votes
1answer
45 views

Galois group of a polynomial over $\mathbb C(t)$

I am learning for my exam tomorrow and I am facing the following task: Compute the Galois group of $x^3-2tx+t$ over $\mathbb C(t)$ At first I want to show that this polynomial is indeed irreducible. ...
2
votes
0answers
31 views

Field extensions and monomorphism

Suppose $[E_1:F]=m<\infty$ and $E_1$ is algebraic extension of $F$. If $K$ is any extension of $F$ then the number of monomorphism of $E_1/F$ into $K/F$ is at most $m$. I am trying to prove this ...
0
votes
2answers
31 views

$[F(a):F] \leq [\mathbb{Q}(a):\mathbb{Q}]$ - proof?

Let $a \in \mathbb{C}$ be algebraic over $\mathbb{Q}$ and $F \subseteq \mathbb{C}$ be a subfield of $\mathbb{C}$. I'm trying to prove that $[F(a):F] \leq [\mathbb{Q}(a):\mathbb{Q}]$ (where $F(a) := ...
0
votes
1answer
22 views

Show that the division algorithm holds for the polynomial ring with integer coefficients when the divisor is of the form g(x) = x-a?

Let $f(x)\in \Bbb Z$ and let $g(x)$ be of the form $g(x) = x-a$. Show that the division algorithm holds in this case. I can see why this holds; it's because $g(x)$ has coefficient one for x so that ...
4
votes
0answers
55 views

Homomorphisms between additive and multiplicative groups of fields

Inspired by this question (In a finite field, is there ever a homomorphism from the additive group to the multiplicative group?), I'm wondering for what fields there exists a non-trivial homomorphism ...
0
votes
0answers
23 views

Show that the polynomial $x^4 + 1$ is reducible in $GF(p)[X]$ for every prime $p$. [duplicate]

Show that the polynomial $x^4 + 1$ is reducible in $GF(p)[X]$ for every prime $p$. This is an excercise from the book: The Linear Algebra a Beginning Graduate Student Should Know; Golan
3
votes
0answers
37 views

Prove that $x^{p^n} -x +1$ is irreducible over $\mathbb{F_p}$ only when n=1 or n=p=2

Question:Prove that $x^{p^n} -x +1$ is irreducible over $\mathbb{F_p}$ only when n=1 or n=p=2 I know it is a duplicate question. However, someone gave some nice hints on this problem and I want to ...
1
vote
1answer
39 views

How to find multiplicative inverses in $\mathbb{Q}[\sqrt[3]{2}]$? [closed]

Is $\mathbb{Q}[\sqrt[3]{2}]=\{ a + b\sqrt[3]{2} + c\sqrt[3]{4} : a,b,c \in \mathbb{Q} \}$ a field? How to find the multiplicative inverse of the above expression $a + b\sqrt[3]{2} + c\sqrt[3]{4}$ ( ...
1
vote
1answer
47 views

Finding a primitive element of $\mathbb{Q}(\alpha,\beta)$ ($\alpha^m=2$, $\beta^n=3$)

I want to prove the following fact: Let $m,n \in \mathbb{N}$ be coprime and $\alpha, \beta \in \mathbb{C}$ with $\alpha^m=2$ and $\beta^n=3$. Then $\alpha\beta$ is a primitive element of ...
4
votes
1answer
33 views

Cyclotomic fields: Determining the fixed field

Let $K=\mathbb Q$ and $L=\mathbb Q(\zeta_8)$, where $\zeta_8$ is a primitve $8th$ root of unity. I have to determine the Galois group of this extension, the subgroups of it and the associated ...
1
vote
1answer
27 views

Question about repeated quotient groups

I was wondering if there was a simpler representation for the quotient group $(K[x,y]/\langle xy\rangle)/\langle x, y-1\rangle$. Where $K[x,y]$ is the polynomial ring with variables $x,y$ over the ...
1
vote
1answer
23 views

why we only need to consider $\lambda(\alpha)$ when we want to write the exact isomorphism?

I am referring to this question Constructing Isomorphism between finite field Consider $\mathbb{F}_3(\alpha)$ where $\alpha^3 - \alpha +1 = 0$ and $\mathbb{F}_3(\beta)$ where $\beta^3 - \beta^2 +1 ...
2
votes
2answers
44 views

Why $s(1-s)$ numbers which are squares in a field are written $\frac{u^2}{1+u^2}$?

Trying to exercise in Math again after years of other activities, I need a little help on this : Let $F$ be some arbitrary field with characteristic > 2. Let $S \subseteq F$ be the set of numbers s ...
0
votes
3answers
43 views

Compute Galois group and minimal polynomial

Let $ \zeta \in \Bbb C$ be a seventh root of $1$. Find the minimal polynomial of the element $ \alpha = \zeta ^{-1} + \zeta$ over $ \Bbb Q$ and show that if $K= \Bbb Q ( \alpha ) $, then $ K/ \Bbb Q$ ...
4
votes
0answers
23 views

Why is $F_5(\root{15}\of t)$ not normal over $F_5(t)$?

(I'm asking this to understand the solution of this question. My idea for a proof so far: $f:=X^{15}-t\in F_5(t)$ is the minimal polynomial of $\root{15}\of t$ in $F_5(t)$, so it suffices to show ...
0
votes
1answer
15 views

Dividing a polynomial in a field $F_5$

I'm dividing $2x^3 + x^2 - 3x + 1$ in $F_5$ with x-3, with long division. I'm not sure if I should be taking modulus at every step? For example, first I multiply (x-3) with $2x^2$, yield $2x^3-6x^2$. ...
4
votes
2answers
58 views

Cyclotomic fields of finite fields

There is a fact that if $K=\mathbb Q$, then the Galois group of the $n$th cyclotomic field over $\mathbb Q$ is isomorphic to $\mathbb Z_n^{*}$. In the case were $K$ is an arbitrary field, we have ...
2
votes
1answer
43 views

Irreducibility of a polynomial with degree $4$

Question: If I want to show irreducibility of $x^4-2x^2+9$ over $\mathbb Q$ Can I do it like this?: I show irreducibility in $\mathbb Z$ because by Gauss the polynomial will be also irreducible in ...
1
vote
2answers
82 views

Showing that $R[X]/(Xf-1) \cong R[1/f]$ [duplicate]

Let $R$ be an integral domain with quotient field $K$. Let $0 \neq f \in R$. I want to prove Statement: $R[X]/(Xf-1) \cong R[1/f]$. Argument: Consider the epimorphism $\phi: R[X] \rightarrow ...
3
votes
1answer
22 views

Formal power series ring, norm. [closed]

Let $k$ be a field. Let $R$ be the formal power series ring $k[[x]]$. Define $N$ on $R \setminus \{0\}$ as follows: $N(f)$ is the smallest $n$ of which the coefficient of $x^n$ in $f$ is nonzero. (a) ...
1
vote
1answer
33 views

For what values of $n > 1$ and a is $x^n - a$ irreducible in $\mathbb Q[x]$?

$$x^n-a$$ So $n$ is any integer greater than 1, and $a$ is any integer. $a$ being any integer is where I am running into trouble. I have already shown and worked out a proof for this being ...
3
votes
1answer
35 views

Finding a subfield of $\mathbb R$ such that $e^2$ is of degree five over that subfield.

My work so far is as follows: \begin{align*} x & = e^2 \\ x^5 & = e^{10} \\ x^5 - e^{10} & = 0. \end{align*} Thus I have a fifth-degree polynomial in $\mathbb R[x]$ with $e^2$ as a zero. ...
1
vote
1answer
44 views

Show that the subset of a real ordered field defined by a ring formula has a least upper bound.

So currently trying to get well practiced in model theory, and i have come across the following question which i need some help with. Esentially let $S \subseteq \mathbb{R}$ be a non empty set, ...
3
votes
2answers
25 views

Determining of the intermediate fields of the $12$th cyclotomic field

Let $\zeta$ be a 12th primitive root over $\mathbb Q$. Determine all intermediate field of $\mathbb Q(\zeta)/\mathbb Q$. My problem is that this is a task from an old exam where you were not allowed ...
3
votes
3answers
88 views

Construction of a field with $8$ elements.

Could someone tell me if one can build a field with $8$ elements?