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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Field reductions. part three

Follow-up to Field reductions. part two For a field $K$, an element $a \in K$, and $K(\setminus a)$ the set of field reductions (maximal subfields of $K$ not containing $a$) are there any interesting ...
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Learning roadmap in Algebra

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
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29 views

Number fields and algebraic integer

Suppose there is a (algebraic) number field $K$. Algebraic integer is a root of some monic polynomial with coefficients in $\mathbb{Z}$. The elements of $K$ are the root of some monic polynomial ...
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51 views

When $\mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha, \beta)$?

Let $\alpha, \beta$ be algebraic numbers over $\mathbb{Q}$. Which necessary and sufficient conditions are known such that $$ \mathbb{Q}(\alpha+\beta)=\mathbb{Q}(\alpha, \beta) \text{ ?} \tag{$\ast$}$$ ...
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If $k$ is a field then $\text{End}_k(k^2)$ is simple

Let $k$ be a field. I have to show that $\text{End}_k(k^2)$ is simple. First of all, I don't see why this is true. For example, if $k=\mathbb{C}(x_1,x_2,\dots)$ then $\varphi\colon k^2\rightarrow ...
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24 views

Regarding nomenclature of a vector related to field automorphisms

Is there a particular designation in use for the following type of vector, constructed by taking a collection of basis elements $\beta_1$, $\beta_2$, $\dotsc$, $\beta_n$ for a field extension, ...
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37 views

Irreducible polynomial

Does there exist an irreducible polynomial over a field K with two roots $a,b$ and $k\in K$ such that $a=b+k$ ? This can't happen if K is of characteristic $0$ , but can it happen if K is of ...
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determine the degree of an extension, check normality

I came across an old exam problem and I wonder if my solution is correct. Let $L=\mathbb{Q}(\omega)$, where $\omega=e^{\frac{2\pi i}{6}}$ is a primitive sixth root of unity: a) determine ...
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Showing the existence of sub fields of a finite field

For each divisor $m$ of $n$, $GF(p^n)$ has a unique sub field of order $p^m$ . The proof of this theorem in Gallian goes like this : Suppose that $m$ divides $n$. Then, since : $(p^n-1) = ...
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51 views

Find a multiplicative inverse of an element in a field

Suppose we have an element $\sigma=p+qa\rho+rd\rho^{-1}\in K$ where $K=\mathbb{Q}(\rho)$ where $[K:\mathbb{Q}]=3$ I want to find a multiplicative inverse of $\sigma$ i .e ...
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29 views

intersection of radical extensions of Q

Are there radical extensions $\mathbb{Q}\subseteq R_1$ and $\mathbb{Q}\subseteq R_2$ such that $R_1 \cap R_2$ is not radical over $\mathbb{Q}$? My guess is that one can find a such example, but I ...
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36 views

Finding the general form of an element in $\frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$

I'm trying to find the general form of elements in the quotient ring: $$R = \frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$$ Now my initial thoughts are to take a general element $f \in R$ so that $f = g + (x^2 ...
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44 views

Prove that every sum of squares in $K$ is a square in $K$, where $K$ is certain field.

Let $K$ be a field such that $f(t)=t^{2}+1$ is an irreducible polynomial in $K[t]$. Let $i$ be a root of $f$ in an algebraic closure of $K$. Suppose every element of $K(i)$ is a square in $K(i)$. ...
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33 views

Splitting field and intermediate fields

Find splitting field $K$ of polynomial $x^3-7$ over $\mathbb{Q} (\sqrt{3} )$ and find $(K: \mathbb{Q} )$ $G(K/ \mathbb{Q} )$ Find intermediate fields between $\mathbb{Q}$ and $K$. So the ...
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35 views

Galois extension of two fields

I am preparing myself to the exam in algebra and I have found the following exercise, which is quite hard for me. Do you have any suggestions for solving it? For fields $K\subseteq L,M\subseteq \bar ...
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24 views

Field extension of fraction field of polynomial ring modulo an ideal.

My apologies for the relatively long question, but I am trying to understand a step in a proof, which needs some preliminary explanation. Let $K$ be a field and $I$ a prime ideal of ...
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39 views

Showing that $\frac{\mathbb{C}[X]}{<x-1>}$ is isomorphic to $\mathbb{C}$

I'm trying to show that $\frac{\mathbb{C}[X]}{<x-1>} \cong \mathbb{C}$ and I am not sure if this argument is correct. Define $\phi: \mathbb{C}[X] \to \mathbb{C}$ by $\sum a_it^i \to \sum a_i$. ...
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Did I Do This Galois Theory Problem Right? Subfields of $\mathbb{Q}(\zeta_{12})$.

Let $\omega$ be a primitive $12$th root of unity. (i) What is $[ \mathbb{Q}(\omega) : \mathbb{Q}]?$ (ii) List the distinct conjugates of $\omega + \omega^{-1}$. (iii) What is $Aut(\mathbb{Q}(\omega ...
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For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \ldots$

For $a=\cos(2\pi/n)$, show that $[\mathbb{Q}(a):\mathbb{Q}] = \{1 \text{ for } n=1,2; (1/2)\phi(n) \text{ for } n>2\}$. $\phi$ is the Euler totient function which gives the number of coprime ...
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34 views

simple extension with algebraic over the field

Assume that $F$ is infinite, that $v,w \in K$ are algebraic over $F$, and that $w$ is a root of a separable polynomial in $F[x]$. How will I be able to prove that $F(v,w)$ is a simple extension of ...
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33 views

Reference for Galois Theory of infinite field extensions.

I would like to ask what is in your opinion the best place to learn about Infinite Galois Theory that requires not much knowledge of topology. I am searching for a text that explains the notions ...
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33 views

Function Field of Degree 3 Ramified at 1 and -1

This question is a homework problem, and I'm having a lot of trouble with it. (a) Determine the number of isomorphism classes of function fields K of degree 3 over $F = \mathbb{C}(t)$ that are ...
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53 views

How many different proper subfields does $K$ have, where $K$ is a field of order$p^n$?

If $p$ is a prime, and $d$ is an integer greater than or equal to 1. Let $n= p^d$, then there exists $K$ being a field of order $p^n$. But how many different proper subfields does $K$ have? The ...
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Topology in Infinite Galois Theory.

I am a final year undergraduate student in Mathematics. I have a good background in algebra up to Galois theory of finite extensions of fields. I have started trying to understand the Galois theory of ...
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60 views

Splitting Field of Irreducible Polynomial over a Finite Field

Suppose $f(x)$ is irreducible over $\mathbb{F}_p[X]$ and let $\alpha$ be a root of $f$ in some extension field. I want to prove the following claim. I have included thoughts below, but I am very ...
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49 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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Why can an ideal generated by a subset be written in this form?

I have a subset $F \subset R$ that generates an ideal $(F)$. Apparently this can be written in the form $$(F)=\{a_1f_1b_1+...+a_kf_kb_k|k \ge 0, f_i \in F, a_i,b_i\in R\}$$ or if $R$ is commutative ...
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40 views

Elementary Field Theory: Extension Field of Degree 2

I'm trying to do/understand the following exercise: "Let $E$ be a finite extension of a field $F$. If $[E:F] = 2$, show that $E$ is a splitting field of $F$."* Background: Just beginning ...
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42 views

A field extension $K \subseteq F$ is finitely-generated if and only if there exists a ring epimorphism $K[t_1 \cdots t_n] \twoheadrightarrow F$

Does this make sense as an alternative definition for a finitely-generated field extension?: A field extension $K \subseteq F$ is finitely-generated if and only if there exists a ring epimorphism ...
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40 views

Composite of two simple extensions

Let $a, b$ be algebraic elements over a field $K$ and suppose at least one of these two elements is separable over $K$. Then, prove that there exists $c$ such that $K(a, b) = K(c)$.
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Factorize polynomial in $\mathbb Z_2[X]$

What is the most efficient way (less time consuming, algorithmically) to factorize polynomials in $\mathbb Z_2[X]$ ? For small degree polynomials, I just try every possibilities (like ...
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70 views

Separability of field extensions

I'm trying to figure out if these statements are equivalent for an arbitrary field extension $L/k$, such that $\text{char} (k) = p$ and $A$ is a reduced commutative $k$-algebra. $1)$ $L/k$ is ...
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41 views

kernel of maps associated to the root of an irreducible polynomial

Let $m(\mu)$ be an irreducible polynomial of degree $d$ over $\mathbb{F}_2$, $F_{2^d} = \mathbb{F}_2[x]/(m(\mu))$ by a field extension given by that polynomial and let $d: \mathbb{F}_2[x] \to ...
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53 views

Can “polar numbers” be added in a sensible way?

Let $\mathbb{P}$ denote the set of all "polar numbers," by which I just mean pairs of real numbers $(r,\theta).$ Note in particular that $r$ is allowed to be negative. Then we can structure ...
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40 views

Direct proof of: $\#($cover of $V) < \#F \; \Rightarrow \;V$ belongs to cover

I'm looking for a direct proof 1 (as opposed to a proof-by-contradiction) of the following theorem: Let $V$ be a vector space over a field $F$ and let $\mathcal{W}$ be a collection of (vector) ...
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40 views

A problem in field norm and division ring

let $D$ be a division ring with center $F$ and let $K$ be a maximal field of $D$. if $N_{D^*}(K*)$ be a maximal on $D^*$, then $K/F$ is galois.
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Extending field automorphisms

Consider an algebraically closed field $K$, and let $F$ be a subfield of $K$. Why every automorphism of $F$ can be extended to an automorphism of $K$? Notice that the field $K$ is not the algebraic ...
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29 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?
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46 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}}) / ...
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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 ...
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Galois extension and subfield

Let $E/F$ be a Galois extension and $Gal(E/F)\cong \Bbb Z/p^3\Bbb Z$. Assume that there exists subfield $K$ of degree $p$ of $E$. (i.e, $[E:K]=p$) Then, show that any proper subfield of $E$ is ...
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Determing the structure of the subgroup of an automorphism group

Suppose we have two automorphisms of an extension field $L=\mathbb{Q}(t)$ for some variable, given by $\sigma: t \mapsto 1-t$ and $\tau : t \mapsto \frac{1}{t}$. Clearly $\langle \sigma , \tau ...
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Independent indeterminate roots and coefficients of a polynomial

Dummit and Foote, Section 14.6: "If the roots of a polynomial $f(x)$ are independent indeterminates over a field $F$, then so are the coefficients of $f(x)$." This is meant to complete the converse of ...
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31 views

How do I prove the multiplicativity of separable degree in general?

Let $K/E/F$ be a tower of algebraic extensions. How do I prove that $[K:E]_s[E:F]_s = [K:F]_s$? This is done in all the books I searched for finite extensions (when it follows trivially from a ...
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64 views

Automorphism of Field Extension Statement

In our Galois Theory course the following statement was given to us in lectures: Suppose that $L : K$ is finite and normal, and $α, β ∈ L$ are roots of an irreducible polynomial $m ∈ K[x]$. Then ...
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77 views

On the fields of rational fractions over $\mathbb{F}_{p}$

Let $K$ be a field with characteristic $p>0$. Let $K^{p}=\left\{ a^{p}:a\in K\right\}$. Prove that $K^p$ is a subfield of $K$. Furthermore, if $K=\mathbb{F}_{p}\left(X\right)$ is the fields of ...
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36 views

Let $F/K$ be a field extension and $a\in F$ such that $[K(a):K]$ is odd

Let $F/K$ be a field extension and $a\in F$ such that $[K(a):K]$ is odd. Prove that $K\left(a\right)=K\left(a^{2}\right)$ and give counterexample if $[K(a):K]$ is even.
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Let $K$ a field with characteristic $p>0$. Show that $\{x \in K : x^{p^n} =x \}$ is a subfield.

Let $K$ a field with characteristic $p>0$. I've shown that for every positive $n$ the set $\{ x^{p^n} : x \in K \}$ is a subfield of $K$, I did this by showing that $F:K\to K: x \mapsto x^{p^n}$ is ...
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35 views

Isomorphism from Adjoining Root of Minimal Polynomial

Let $L$ be a field extension of $K$, and let $m_{\theta}(x)\in K[x]$ be the minimal polynomial of $\theta\in L$. How can we show that $K[\theta]\cong K[x]/m_{\theta}(x)$? I couldn't help but notice ...
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81 views

Reducibility of $x^6+x^3+1$ over various fields

I'm working through a (non-examined) question sheet and have this problem. Is $x^6+x^3+1$ irreducible over the following fields; (I) $\mathbb{F}_2$, (II) $\mathbb{F}_3$, (III) $\mathbb{F}_{19}$ ...