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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countable set that contains 1 and pi and has polynomial with coefficients in set s.t. all real roots are in set

Deduce that there is a countable set X that contains the real numbers 1 and pi and has the further property that if P is any non-zero polynomial with coefficients in X, then all real roots of P belong ...
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46 views

An irreducible polynomial cannot share a root with a polynomial without dividing it

There is a lemma of Galois stating, "An irreducible equation can have no common root with a rational equation without dividing it". His definitions are a little bit imprecise, but I think he means: ...
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64 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 ...
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1answer
35 views

The Galois closure

If $\Bbb K$ is an extension of $\Bbb Q$ having degree 4, why is the Galois group corresponding to the Galois closure of $\Bbb K$ a subgroup of $S_4$?
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1answer
207 views

Capelli Lemma for polynomials

I have seen this lemma given without proof in some articles (see example here), and I guess it is well known, but I couldn't find an online reference for a proof. It states like this: Let $K$ be ...
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1answer
31 views

Proving the Galois Group of an extension is abelian

Let $E_{1}, E_{2}$ be subfields of $\mathbb{C}$. Suppose $E_{1}|\mathbb{Q}$ and $E_{2}|\mathbb{Q}$ are finite Galois extensions and $G(E_{1}:\mathbb{Q})\cong$ $\mathbb{Z}_{6}\cong$ ...
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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 ...
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Field Extensions Identities

I'm working on proving some identities but I need some help clarifying the notation and what exactly each statement is saying. Prove the following identities. (a) $K(A) = QF (K[A])$ (b) $R[A_1 ...
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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 ...
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1answer
36 views

Galois field of order $p^n$

Can someone help me in establishing an image of how the group looks like. I am having a hard time visualizing it.
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Splitting Fields Proof

Let $K\subseteq L$ be fields and $ f \in K[X] \setminus K$. Prove that the following are equivalent. (a) $L$ is a splitting field for $f$ over $K$ (b) $f$ splits over $L$ and $L=K(a_1,\ldots,a_n)$ ...
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1answer
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If an identity in the language of rings holds for all fields, does it necessarily hold for all commutative rings?

It is weirdly difficult to find new identities for ring theory (other than commutativity) that make it more like field theory. This motivates my: Question. If an identity in the language of rings ...
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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 ...
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1answer
38 views

$x^4-3x^2+4$ irreducible over over $\mathbb{Q}$

I need to prove irreducibility of $x^4-3x^2+4$ over $\mathbb{Q}$. It can't have any linear factor since it doesn't have any root in $\mathbb{Q}$ because any $\alpha \in \mathbb{Q}$ is a root only ...
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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 ...
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3answers
33 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 ...
2
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1answer
31 views

Generators of $\mathbb{F}_9/\mathbb{F}_3$ that do not generate $\mathbb{F}_9^{\times}$

Find a generator of the extension $\mathbb{F}_9/\mathbb{F}_3$ that does not generate the multiplicative group $\mathbb{F}_9^{\times}$. how many such elements exist? what are their minimal ...
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2answers
116 views

$x^5-x^2+1$ is separable over all fields

Prove that $p(x)=x^5-x^2+1$ is separable over all fields. When the field is finite or of characteristic zero it is automatically true, since any polynomial is separable. The definition of ...
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2answers
115 views

The angle $168^\circ$ is constructible

Prove that the angle $\theta=168^\circ$ is constructible using a straightedge and a compass. It is enough to show that the number $\cos\theta$ is constructible, and WolframAlpha gave ...
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1answer
40 views

find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
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3answers
89 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 $F(a)$ ...
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72 views

Non-cyclic finite extensions of fixed fields of infinite order automorphisms of non-algebraically closed fields

This is a problem from Galois theory: Suppose that $F$ is algebraically closed, $\lambda$ is an automorphism of infinite order, and $f=F^\lambda$. Show that any finite extension of $f$ is cyclic. ...
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1answer
80 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$$ F_m:=\prod \left\lbrace f \in \mathbb{F}_q[X]; f \ \text{irreducible, nomic}, \text{deg}(f)=m \right\rbrace $$ ...
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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 ...
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2answers
36 views

Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5

Find amount of invertible matrices of size $3 \times 3$ over residue field modulo 5. I will just add that this task is slightly ahead of my knowledge of field theory. So any pointers would be ...
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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?
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1answer
38 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 ...
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3answers
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$\mathbb Q[x]/\langle x^2+1\rangle$ 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 ...
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1answer
51 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 ...
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1answer
33 views

Simplifying presentation of elements of finite field

Let me describe my question through an example. Finite field of order (for example) 8 can be constructed as $\mathbb{F}_8 = \mathbb{F}_2[t]/(t^3 + t + 1)$. So one of a natural presentation of the ...
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1answer
79 views

Why characters are continuous

According to Wikipedia: ''Every character is automatically continuous from $A$  to $\mathbb C$, since the kernel of a character is a maximal ideal, which is closed. '' where $A$ is a Banach algebra. ...
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How many basis are in n-dimensional vector space over field of q-elements

How many basis are in n-dimensional vector space over field of q-elements? Thanks in advance! Any help is appreciated
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1answer
18 views

Why does the 2-D vector space would imply the splitting field?

Why does the 2-D vector space would imply the splitting field?
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1answer
62 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 ...
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2answers
33 views

Shouldn't the induction hypothesis be taken only on $n?$

I'm having problem in getting the underlined statement from Gallian text: Shouldn't the induction hypothesis be taken only on $n?$ But here the author also assumed the case for arbitrary field in ...
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2answers
796 views

Grassmann numbers as eigenvalues of nilpotent operators?

The following question is motivated by the construction of the fermionic path/field integral, as done for example in Altland & Simons "Condensed Matter Field Theory". Consider the vector space ...
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1answer
21 views

How to show that $\mathbb Q[\sqrt 2]$ is the smallest subfield of $\mathbb R$ containing $\mathbb Q$ and $\sqrt 2?$

How to show that $\mathbb Q[\sqrt 2]=\{a+b\sqrt 2:a,b\in\mathbb Q\}$ is the smallest subfield of $\mathbb R$ containing $\mathbb Q$ and $\sqrt 2?$ I've shown that $\mathbb Q[\sqrt 2]$ is a subfield ...
0
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1answer
21 views

Is it jusfied then to say $\mathbb R$ the splitting field of $x^2-1$ over $\mathbb R?$

Gallian text define splitting field as: Take for example $x^2-1\in\mathbb R[x]$ Due to the definition $x^2-1$ splits over $\mathbb R$ as well as over $\mathbb Q,$ a proper subfield of $\mathbb R.$ ...
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3answers
19 views

Shouldn't the yellow marked $a_0$ be $a_0+\langle p(x)\rangle?$

I'm having problem in getting the proof from Gallian text in the higlighted region: Shouldn't the yellow marked $a_0$ be $a_0+\langle p(x)\rangle?$ Edited: Shouldn't the $a_i$'s in the ...
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1answer
33 views

Concerning a Cyclic Galois Group

Why is it that: $\forall K \supseteq \mathbb{Q}(\mathbb{i}), G=Gal(K / \mathbb{Q}) = \langle \sigma \rangle \implies \sigma (\mathbb{i}) = - \mathbb{i}$? (Note: I am guessing that $\sigma ≠ ...
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Notational issue

Let $K = F(t)$. If $r \in K: (\nexists c \in F: r(t) = c \forall t)$ is a rational function and $L = F(r(t))$, then what form does $f \in L$ have? Is it a rational function where the coefficients are ...
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2answers
48 views

Show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois; prove that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}(2^{1/2})$ is Galois

I would like to show that $\mathbb{Q}(2^{1/4}) / \mathbb{Q}$ is not Galois. Can I just say that it is not separable because $2^{1/4} \in \mathbb{Q}(2^{1/4})$ but its minimal polynomial in $\mathbb{Q}$ ...
0
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1answer
26 views

$\mathbb R[x]/\langle x^2+1\rangle\simeq\mathbb C$ and $\mathbb R[x]/\langle x^2+1\rangle$ contains a zero of $x^2+1$

How to show that $\mathbb R[x]/\langle x^2+1\rangle\simeq\mathbb C$ and $\mathbb R[x]/\langle x^2+1\rangle$ contains a zero of $x^2+1$ Due to the division algorithm, $\mathbb R[x]/\langle ...
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1answer
16 views

Permutation of a fixed field is an intermediate field corresponding with the conjugate of the group corresponding to the fixed field

The following is my question: Let $K/F$ be a Galois extension with Galois group $G = Gal(K/F)$, with intermediate field $L: F \subseteq L \subseteq K$ which corresponds to subgroup $H \leq G$ by the ...
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1answer
30 views

Every Intermediate Field of Abelian Galois Field Extension is Splitting Field of a Separable Polynomial

This is my question: Suppose the $K/F$ is a Galois extension with an abelian Galois group $G$. Prove that every intermediate field $L: F \subseteq L \subseteq K$ is the splitting field (over $F$) of ...
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0answers
32 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 ...
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2answers
40 views

Prove the ring $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ has inverse and is a field

How can I prove that $\frac{1}{a+b\sqrt[3]{2}+c\sqrt[3]{4}}$ is of the form $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ (i.e. that $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ is a field) for all rational $a,b,c$ and ...
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2answers
37 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 ...
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1answer
47 views

Algorithm to find representation of an element of field extension $\mathbb{Q}(q)$ in the form $\sum a_i q^i$

Let $\mathbb{Q}(q)$ be a field extension of $\mathbb{Q}$, where $q$ is a real root of some monic irreducible polynomial $p(x) \in \mathbb{Z}[x]$ of degree $d=3$. Given $x \in \mathbb{R}$, (or ...
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2answers
40 views

Constructing expressions

Suppose you are given all the elementary symmetric functions of $n$ variables $x_1,x_2,...,x_n$ and two rational functions $A(x_1,x_2,...,x_n)$ and $B(x_1,x_2,...,x_n)$ in the same $n$ variables that ...