0
votes
1answer
10 views

Fields with Additive identity powers

Would it be possible to have a field (or field-like structure) with an additive identity $k$ where $k^a\neq k^b$ for $a\neq b$? I need this because I'm working with a field-like structure where if I ...
2
votes
1answer
47 views

Subfield of the Galois Group of $x^5 - 1$

Why is it that the subfield fixed by the subgroup of this Galois group is $\mathbb{Q}(\sqrt5)$. Can someone explain it without using the cyclotomic extension of $\mathbb{Q}$? Thank you edit: Using ...
0
votes
1answer
15 views

Separable and Splitting fields

Does a separable field imply splitting field? I am thinking yes. For $F < E$ If every $\alpha \in E$ is separable over $F$ then it must be that $E$ is a $S.F.$ over $F$? Also we have the ...
0
votes
1answer
16 views

In algebraic extension, field homomorphism induces isomorphism.

I read this page's first answer. But I'm curious about why $\varphi$ induces injective map $S \to S$. Isn't it possible to make $\varphi(\alpha)= k$ such that $k$ is not root of $f(X)$?
3
votes
0answers
45 views

Equivalence of definitions for “normal extension” and how to lift isomorphisms to them

Briefly: I want to prove that these two definitions for "normal extension" are equivalent: "$K$ is a splitting field for a collection of polynomials in $F[x]$" vs. "Every irreducible polynomial in ...
1
vote
1answer
41 views

Automorphism that maps primitive roots of unity.

Let $ w_1,...,w_{ \phi(n)}$ be the primitive $n$th roots of unity of $ t^n -1 \in \mathbb Q[t]$. Show that for each $ 1 \le i \le \phi (n)$, there exists an $ \sigma\in Aut \mathbb Q(w_1)$ satisfies $ ...
0
votes
1answer
58 views

Field of algebraic numbers over $\mathbb{Q}$

Let $F$ be the field of algebraic numbers over $\mathbb{Q}$. I do remember that this means $F$ is a field extension of rationals over $\mathbb{Q}$. How do I show that the field extension of $F$ is ...
0
votes
1answer
30 views

What exactly are the elements of $\mathbb{Z}_p[x]/\langle p(x) \rangle$?

It is wellknown that for a polynomial ring $\mathbb{Z}_p[x]$, $\mathbb{Z}_p[x]/\langle p(x) \rangle$ for prime $p$ is a field if and only if $p(x)$ is irreducible over the given polynomial ring, in ...
0
votes
1answer
53 views

Basis for $\mathbb{Q}[\sqrt{8}]$ over $\mathbb{Q}[\sqrt{2}]$

Provided that $x^2-8$ is the minimal polynomial for $\mathbb Q[\sqrt8]$ and $x^2-2$ is minimal for $\mathbb Q[\sqrt 2]$ we should have a basis with four elements. Thus far I know $1$ and $\sqrt 2$ ...
2
votes
1answer
78 views

Is $x^4+2$ irreducible over $\Bbb{Q}(i)$?

Let $f(x)=x^4+2$. Using the Eisenstein test to $f(x+2)$, one can show that $f$ is irreducible over ${\Bbb Q}$. Let $\beta$ be a complex root of $f$. Then the question in the title is equivalent to ...
1
vote
3answers
51 views

Dummit and Foote page 526

I'm having trouble with a line of example 2 on page 526. Consider the field $\mathbb{Q}(\sqrt{2},\sqrt{3})$. generated over $\mathbb{Q}$ by $\sqrt{2}$ and $\sqrt{3}$. Since $\sqrt{3}$ is of degree ...
1
vote
1answer
53 views

Dummit and Foote page 512 claim

Dummit and Foote Abstract Algebra page 512 Given any field F and any polynomial $p(x)\in F[x]$ one can ask a similar question: does there exist an extension K of F containing a solution of the ...
3
votes
2answers
60 views

Is the algebraic subextension of a finitely generated field extension finitely generated?

This question is motivated by this other question (and its answer). Suppose we have a field $F$, possibly imperfect. Consider the finitely generated field extension $F(a_1,\ldots,a_n)$. Is it always ...
1
vote
1answer
20 views

Binomial formula over an arbitrary field

I'm working on a problem (namely, if $\alpha + \beta$ is algebraic over $F$ then $\alpha$ is algebraic over $F[\beta]$), and the binomial formula appeared. For the problem, I used the fact that, for ...
5
votes
1answer
59 views

Problem of Galois Extension

$\Bbb K$ is a non-Galois extension of $\Bbb Q$ and $[K:\Bbb Q]=4$. If $\Bbb F$ is the Galois closure of $\Bbb K$ then show that $Gal(\Bbb F/\Bbb Q)$ is either $S_4, A_4$ or $D_8$ with order 8. ...
3
votes
2answers
51 views

Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
5
votes
2answers
63 views

Is there always an intermediate field between non-prime field extension?

If we have $[E:F]=n$, where $n$ is not a prime number but is finite, can we like prime factorize $n=p_1p_2...p_r$, so that we have $[E:F]=[E:K_1][K_1:K_2]...[K_{r-1}:F]$ and each of the ...
3
votes
1answer
58 views

Problem about Galois Extension.

$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
0
votes
1answer
39 views

A question on solvable groups.

Let's suppose $S$ is a group and can be written as $S= G_1\times G_2\times\cdots\times G_K$; here each $G_i$ is a group of prime power order. And $o(S)= n.$ Is this $S$ solvable group? If yes how will ...
0
votes
2answers
28 views

How many elements does $\mathbb F = \mathbb Z_{7}[x]/I$ contain?

Let $p(x) \in \mathbb Z_{7}[x]$, given by $p(x) = x^{2}+3x+1$ and let $I = <p(x)>$ be the ideal in $\mathbb Z_{7}[x]$ constructed by $p(x)$. How many elements does $\mathbb F = \mathbb ...
0
votes
1answer
37 views

Let $\mathbb F = \mathbb Z_{3}[x]/\langle x^{2}+1 \rangle$ Show that $\phi : G \to \mathbb F\backslash\{0\}$ defined by …

Let $G = \left\{ \begin{bmatrix} \alpha & \beta \\ 2\beta & \alpha \end{bmatrix} \;\Bigg| \; \alpha,\beta \in \mathbb Z_{3} (\alpha,\beta) \neq (0,0) \right\}$ Let $\mathbb F = \mathbb ...
2
votes
1answer
35 views

Order Of The Intersection of Two Subfields.

Last question haha, Let $E$ and $F$ be subfields of $GF(p^n)$. If $|E| = p^r$ and $|F| = p^s$, what is the order of $E\cap F$? I read a corollary that "A finite field of order $p^n$ contains a ...
2
votes
3answers
32 views

Showing an element in a Finite Field can be written as a power.

I had a question that I'm stuck with: Show that every element in $GF(p^n)$ can be written in the form of $a^p$ for some unique $a\in GF(p^n)$. So this field is the splitting field for the polynomial ...
0
votes
3answers
82 views

showing $Q[\sqrt 2] = Q(\sqrt 2)$

The question came in my exam. $Q[\sqrt 2] = \{ a + b \sqrt2 \;| a,b \in Q\}$ and $Q(\sqrt 2)$ is minimal subfield of it's extension containing $Q$ and $\sqrt 2$. (In my book) It calls $F(a)$ ...
4
votes
1answer
41 views

Multiplicative order in field extension

Let $F/K$ be some field extension (both are finite fields) and $u$ be some element in $F$. I want to know if $u^{|K|} = u$ implies $u \in K$. And why?
0
votes
1answer
34 views

Proving that a function has no repeated roots

Let $\mathbb{Q} \subset F$ be a field extension. Prove that if $f(x) \in F[x]$ is irreducible, then it has no repeated roots in any field extension of F. as a hint we were given that a repeated root ...
0
votes
0answers
38 views

Notational Clarification - Abstract Algebra

I'm going through a paper on homomorphic encryption by Smart and Vercauteren entitled "Fully Homomorphic SIMD operations" and had a question about some notation used in the paper. In section 2 of the ...
5
votes
1answer
78 views

Difficult algebraic problem - irreducible polynomials

Let $p$ be a prime number and $q=p^n$ with $n \in \mathbb{N}$. We define the polynomial$$ F_m:=\prod \left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, nomic}, \text{deg}(f)=m \right\rbrace $$ ...
1
vote
3answers
45 views

$\mathbb Q[x]/\langle x^2+1\rangle$ is a splitting field of $x^2+1$ over $\mathbb Q.$

Due to the Kronecker's theorem, $x^2+1\in\mathbb Q[x]$ splits over $\mathbb Q[x]/\langle x^2+1\rangle.$ But how to show that $\mathbb Q[x]/\langle x^2+1\rangle$ is a splitting field of $x^2+1$ over ...
0
votes
1answer
49 views

Order of element of multiplicative group of finite field mod polynomial

If $K$ is a finite field of size $q$ and $f$ is a degree $n$ polynomial in $K[x]$, then we can form the quotient field by modding out this polynomial. Elements of this quotient field are of the form ...
1
vote
0answers
31 views

Union of field extensions over Q

I am asked to prove that $L=\bigcup_{n=1}^\infty\mathbb{Q}(\sqrt[n]2)$ is an algebraic field extension over $\mathbb{Q}$. So far I have: Let $\beta\in L$, then by definition of union there exists a ...
1
vote
2answers
70 views

When does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$?

Let $p$ be a prime integer, and let $q=p^r$ and $q'=p^k$. For which values of $r$ and $k$ does $x^{q'}-x$ divide $x^q-x$ in $\mathbb{Z}[x]$? From Artin's Algebra, Chapter 15, problem 7.12 from the ...
2
votes
1answer
65 views

Why $K(u)$ is a field?

Let $F$ be an extension field of $K$ and $u\in F$. How do we know that adjoining an element of F to K, makes $K(u)$ a field? I know that $Q(\sqrt2)=\{a+b\sqrt2|a,b\in Q\}$ is a field, but in the ...
0
votes
3answers
59 views

Proof about finite dimensional vector spaces over fields

Prove that every finite dimensional vector space $V $of dimension $n$ over a field $F$ is isomorphic to the vector space $F^n$. Okay, lot's of stuff here. I think most of the reason I can not do this ...
2
votes
2answers
41 views

Ring homomorphism with field as image, is the pre-image also a field?

Let $f:R\rightarrow S$ be a surjective ring homomorphism. $R,S$ are both integral domains. Suppose $S$ is a field, then is $R$ also a field? A possible useful fact: A finite integral domain is a ...
1
vote
2answers
35 views

Analogy between the quaternion ring and extensions of the rationals

I've started studing fields and their extensions. As an exercise I proved that $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$ by showing that $B=(1,\sqrt2,\sqrt3,\sqrt6)$ is a base for the extension field ...
2
votes
2answers
78 views

Field extension of a finite field

Let $E$ be an extension field of a finite field $F$ , where $F$ has $q$ elements. Let $a \in E$ be algebraic over $F$ of degree $n$. Prove that $F(a)$ has $q^n$ elements. I am not sure how to do this ...
0
votes
1answer
33 views

Extension Fields and Quotients

In the Dummit and Foote 3ed chapter on field extensions (ch. 13), it is stated as a theorem (6) that $ F(\alpha) \cong F[x]/(p(x))$ where $\alpha$ is a root of $p(x)$ and goes on to state that any ...
3
votes
2answers
94 views

Field extensions and algebraic/transcendental elements

Let $E$ be an extension of field $F$, and let $\alpha, \beta \in E$. Suppose $\alpha$ is transcendental over $F$ but algebraic over $F(\beta)$. Show that $\beta$ is algebraic over $F(\alpha)$. ...
1
vote
1answer
33 views

Splitting field in $\mathbb{C}$ over $\mathbb{Q}$

I want to find the splitting field in$\mathbb{C}$ of $x^4-4$ over $\mathbb{Q}$. I solved for the zeros, which is $i\sqrt2, -i\sqrt2, \sqrt2, -\sqrt2$, so the splitting field, say $E$, is just ...
8
votes
2answers
160 views

$\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}$?

Is there an easy way to see that $$\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})=\mathbb{Q}?$$ I know that $\mathbb{Q}(\sqrt[3]{2})\cap\mathbb{Q}(\sqrt[3]{5})$ is a subfield of ...
1
vote
1answer
42 views

Computing a field extension by hand

Let $k$ be a field, and $K=k(t)$ be the field of rational functions (where $t$ is indeterminate). Let $F=k(t^2)$. A typical element of $F$ will look like: $$ \frac{\displaystyle \sum_{i=0}^{n} a_{2i} ...
5
votes
0answers
56 views

The reducibility of polynomial $x^{2^n}+x+1$ over $Z_2$ [duplicate]

I meet the problem in Hungerford's Algebra, page 282 as follows: If $n>2$, then the polynomial $x^{2^n}+x+1$ is reducible over $Z_2$ . I have no any idea on this. But I really want to know how to ...
1
vote
1answer
34 views

Splitting field in multiple field extension

Let $F\subset K\subset E$ be fields, $f(x)\in F[x]$. E is the splitting field of $f(x)$ over $F$. Prove $E$ is the splitting field of $f(x)$ over $K$. My problem with this proof is that I do not know ...
5
votes
0answers
86 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
1
vote
1answer
61 views

Splitting field of a set of separable polynomials implies separability of extension.

Let $F$ be a splitting field of $S\subset K[x]$ over $K$, where $S$ is a set of separable polynomials. I want to show that $F$ is separable over $K$, meaning for all $u\in F-K$, the irreducible ...
0
votes
0answers
17 views

Application of the Isomorphism Extension Thm

Let $K$ be an algebraic closed field. Show that every isomorphism $\sigma$ of $K$ onto a subfield of itself $s.t. \ K$ is alg. $/ \sigma[K]$ is an automorphism of $K$? this seems too clear to me that ...
0
votes
0answers
21 views

For which $d\in\mathbb{Z}$, $\mathbb{Q}(\sqrt{d})$ primitive root of unity of order $p>2$ prime

If $p>2$ is a prime number, then I have to find $d\in\mathbb{Z}$ such that we have a primitive root of unity of order $p$. I know that $d<0$ because otherwise, you can never have a root of ...
1
vote
1answer
29 views

Why is every monomorphism of E into $\overline{\mathbb{Q}}$ a $\mathbb Q$-monomorphism of E into $\overline{\mathbb{Q}}$?

I've been trying to solve the problem below, but I'm not even sure how to get started. Any help would be greatly appreciated. I feel like there is a key insight that will solve the problem, but I'm ...
5
votes
1answer
141 views

Show from the axioms: Addition in a quasifield is abelian

According to wikipedia a quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is a group. (As usual, we denote its identity element by $0$.) $(Q\setminus\{0\},\cdot)$ is a loop. (Its ...