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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1answer
97 views

If $K,E$ are subfields of $\Omega/F$ then $KE/F$ is a finite Galois imply $K/K\cap E$ is Galois?

Let $\Omega/F$ be a field extension and $K,E$ be two subfields of $\Omega/F$. Assume that $KE/F$ is a finite Galois. I have a theorem in my lecture notes that claim $\text{Gal}(KE/E)\cong ...
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4answers
84 views

If $K/F$ is Galois and $E$ is a subextension then $E$ is generated by roots of a polynomial over $F$?

Let $K/F$ be finite Galois field extension, then $K$ is the splitting field of a separable polynomial $p$ over $F$, i.e. $K=F(a_{1},..a_{n})$ where $p=(x-a_{1})...(x-a_{n})$. My question is: is it ...
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2answers
5k views

Galois Field GF(4)

Question: Why is the table of GF(4) look like the one below? I know it has to do with the fact that 4 is composite Let GF(4) = {0,1,B,D} Addition: $$ \begin{array}{c|cccc} + & 0& 1& ...
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1answer
221 views

Finding quadratic extensions in $\mathbb{Q}\left(\sqrt{i+2}\right)$

Let $\alpha=\sqrt{i+2}$ and let $F=\mathbb{Q}\left(\alpha\right)$. Note that $\left[F:\mathbb{Q}\right]=4$ since the minimal polynomial of $\alpha$ over $\mathbb{Q}$ is $x^{4}-4x^{2}+5$. Show ...
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2answers
431 views

A question about a proof of a weak form of Hilbert's Nullstellensatz

I'm trying to prove the following (corollary 5.24 page 67 in Atiyah-Macdonald): Let $k$ be a field and let $B$ be a field that is a finitely generated $k$-algebra, i.e. there is a ring homomorphism ...
3
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2answers
486 views

A question about a weak form of Hilbert's Nullstellensatz

Corollary 5.24 on page 67 in Atiyah-Macdonald reads as follows: Let $k$ be a field and $B$ a finitely generated $k$-algebra. If $B$ is a field then it is a finite algebraic extension of $k$. We know ...
0
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1answer
64 views

Proving that $\Phi_{n}$ is irreducible (a problem with the proof)

I am trying to follow the proof in the book Abstract Algebra by Dummit and Foote (Theorem 41, pg. 554) that $\Phi_n$ is an irreducible monic polynomial in $\mathbb{Z}[x]$ of degree $\varphi(n)$. What ...
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1answer
59 views

Why $g(x^{p})=(g(x))^{p}$ in the reduction mod $p$?

In one of the proof in the book "Abstract Algebra'' by Dummit and Foote (Theorem 41, pg. 554) we have a monic polynomial $g(x)\in\mathbb{Z}[x]$, and the book claims that $g(x^{p})=(g(x))^{p}\mod p$ ...
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3answers
218 views

$x$ algebraic over $K$, $v$ a polynomial in $x$ then $v$ algebraic?

In the proof of proposition 5.23 Atiyah-Macdonald on page 66 use that if $x$ is algebraic over $K$ and $v = a_n x^n + \dots + a_1 x + a_0$ then $v$ is algebraic over $K$ (where $K$ is the field of ...
4
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2answers
146 views

Proving that if $\mathrm{char}(F)=p>0$ then if $g(x)\in F[x]$ is irreducible then $g(x)$ have multiple roots iff $g'(x)=0$

I am going over my lecture notes in my Field theory class and I saw this following statement without a proof: if $\mathrm{char}(F)=p>0$ then if $g(x)\in F[x]$ is irreducible then $g(x)$ have ...
4
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1answer
215 views

Showing existence of a field extension of degree $n$ for a finite field $F$

EDIT: Just mentioning that this is a homework question. This is my first time posting a question on math.stackexchange, so I hope you find it in your hearts to forgive any stylistic or rule ...
3
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1answer
169 views

Additive inverse

Let $F$ be the set of $\alpha\subset \mathbb{Q}$ with following properties. (I) $\alpha ≠ \emptyset$ and $\alpha ≠ \mathbb{Q}$ (II) $p\in \alpha$ and $q<p$ ⇒ $q\in \alpha$ (Notice that it's ...
2
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1answer
128 views

If the polynomial $f$ is zero on the nonzero set of another polynomial $g$, does $f=0$?

Suppose $f(x_1,\dots,x_n)$ is a polynomial in $n$ indeterminates over an infinite field $F$. Suppose $f((a_i))=0$ for all $n$-tuples $(a_i)$ such that $g((a_i))\neq 0$, where $g(x_1,\dots,x_n)$ is ...
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2answers
936 views

Why algebraic closures?

Let me begin by summarizing the question: Why do we care about fields closed under rational exponentiation, and less about fields closed under other operations? Historically the solution for ...
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2answers
130 views

Field of algebraic reals over the rationals

Let $L$ be the subfield of $\mathbb{R}$, of all reals that are algebraic over $\mathbb{Q}$: $L = \{ x\in \mathbb{R} : x \text{ is algebraic over } \mathbb{Q} \}, \;\;\; \mathbb{Q} \subseteq L$. Let ...
8
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1answer
390 views

Problem 18.7 in I. Martin Isaacs' Algebra

I am trying to solve the following problem in I. Martin Isaacs' Algebra: A graduate course, p.290: Let $f(X),g(X) \in F[X]$ and suppose $E \supseteq F$ is the splitting field both for $f(X)$ and ...
2
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1answer
91 views

Is there a necessary and sufficient condition to determine the generators of $\mathbb{Z}_p^\times$?

This is something I was wondering about. I know that the generators of the cyclic groups $(\mathbb{Z}_n,+)$ are precisely those integers coprime to $n$, and there are $\phi(n)$ of them. Now the ...
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4answers
868 views

Why is the product of all units of a finite field equal to $-1$?

Suppose $F=\{0,a_1,\dots,a_{q-1}\}$ is a finite field with $q=p^n$ elements. I'm curious, why is the product of all elements of $F^\ast$ equal to $-1$? I know that $F^\ast$ is cyclic, say generated by ...
3
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1answer
169 views

Problem 18.1 in I. Martin Isaacs' Algebra

I am trying to prove the following: Let $E/F$ be an arbitrary extensions. Show that $E/F$ is normal if and only if $E$ is the union of all those intermediate fields $K$ such that $K$ is the splitting ...
3
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1answer
326 views

Two notions of uniformizer

Let $X$ be a projective algebraic curve and consider a `uniformizing' map $h:X \rightarrow \mathbb{P}^1$. Is there any connection between this notion of uniformizer and a uniformizer of the maximal ...
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3answers
270 views

When is $\mathbb{F}_p[x]/(x^2-2)\simeq\mathbb{F}_p[x]/(x^2-3)$ for small primes?

I've been considering the rings $R_1=\mathbb{F}_p[x]/(x^2-2)$ and $R_2=\mathbb{F}_p[x]/(x^2-3)$, where $\mathbb{F}_p=\mathbb{Z}/(p)$. I'm trying to figure out if they're isomorphic (as rings I ...
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3answers
112 views

Is there a good way to solve for the inverse of $(u^2-u+4)$?

I'm having trouble calculating the inverse of a polynomial. Consider the polynomial $f(x)=x^3+3x-2$, which is irreducible over $\mathbb{Q}$, as it has no rational roots. So $\mathbb{Q}[x]/(f(x))\simeq ...
3
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1answer
309 views

If $F$ is a formally real field then is $F(\alpha)$ formally real?

Let us call a field $F$ $\textit{ formally real }$ if $-1$ is not expressible as a sum of squares in $F$. Now suppose $F$ is a formally real field and $f(x)\in F[x]$ be an irreducible polynomial of ...
4
votes
1answer
785 views

Finding a primitive element for the field extension $\mathbb{Q}(\sqrt{p_{1}},\sqrt{p_{2}},\ldots,\sqrt{p_{n}})/\mathbb{Q}$

Let $p_1,\ldots,p_n\in\mathbb{N}$ be different prime numbers, it can be shown that $[\mathbb{Q}(\sqrt{p_1},\ldots,\sqrt{p_n}):\mathbb{Q}]=2^n$ and in any case it is clearly finite since ...
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3answers
875 views

How to think of the field $F(\alpha)$

The way I learned it was given a field extension $F \subset E$, and an element $\alpha \in E$ $$F(\alpha) := \{p(\alpha)/q(\alpha) : p(x), q(x) \in F[x] ,q(\alpha) \not = 0\} $$ Is there an easier ...
2
votes
1answer
86 views

Splitting field and subextension

Definition: Let $K/F$ be a field extension and let $p(x)\in F[x]$, we say that $K$ is splitting field of $p$ over $F$ if $p$ splits in $K$ and $K$ is generated by $p$'s roots; i.e. if ...
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2answers
400 views

If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$

If $E$ and $F$ are subfields of a finite field $K$ and $E\cong F$, prove that $E = F$. A finite field is a simple extension of each of its subfields and $\mathbb{Z}_p$ is a subfield of every finite ...
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1answer
133 views

Normal Field Extension

$X^4 -4$ has a root in $\Bbb Q(2^{1/2})$ but does not split in $\Bbb Q(2^{1/2})$ implying that $\Bbb Q(2^{1/2})$ is not a normal extension of $\Bbb Q$ according to most definitions. But $\Bbb ...
2
votes
2answers
116 views

$f$ is irreducible in $\Bbb F[x]$

Let $\Bbb F$ be a field of characteristic $p\gt 0$ and $f(x)=x^{p^n}-c \in\Bbb F[x]$ where $n$ is a positive integer. If $c \notin \{a^p:a\in \Bbb F \}$, show that $f$ is irreducible in $\Bbb ...
2
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1answer
195 views

A property equivalent to $K$ being a finite-dimensional, normal extension field of $F$

Prove that a finite-dimensional extension field $K$ of $F$ is normal if and only if it has this property: Whenever $L$ is an extension field of $K$ and $\sigma :K\rightarrow L$ an injective ...
2
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1answer
152 views

Number of elements in $\mathbb{Z}_p[x]/ \langle f \rangle$

I want to determine the number of elements in $\mathbb{Z}_p[x]/ \langle f \rangle$ where $f \in \mathbb{Z}_p[x]$ is an irreducible polynomial with $k$ degree bigger than 2. Is the number of elements ...
28
votes
1answer
2k views

Is there a purely algebraic proof of the Fundamental Theorem of Algebra?

Among the many techniques available at our disposal to prove FTA, is there any purely algebraic proof of the theorem? That seems reasonably unexpected, because somehow or the other we are depending ...
3
votes
3answers
888 views

Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$

Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$. Ok so originally I messed around with $x^3 + x +1$ for a bit looking for an easy way to factor it and eventually decided that the ...
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2answers
2k views

Is vector space a field? Or more than that?

As you all know, vector space is closed under scalar multiplication, scalar product, vector product and addition. If I take scalar product, vector space is a field, but if i take vector product, ...
6
votes
1answer
607 views

Calculating the norm of an element in a field extension.

Given a number field $\mathbb{Q}[\beta]$, where the minimal polynomial of $\beta$ in $\mathbb[Z][x]$ has degree $n$, I would like to calculate the norm of the general element ...
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2answers
2k views

Constructing a Galois extension field with Galois group $S_n$

Constructing a Galois extension field $E$ with $Gal(E/F)= S_n$ How do I construct one?
4
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2answers
648 views

If $K$ is an extension field of $\mathbb{Q}$ such that $[K:\mathbb{Q}]=2$, prove that $K=\mathbb{Q}(\sqrt{d})$ for some square free integer $d$

I think I have the later parts of this proof worked out pretty well but what's really stumping me is how to go from knowing $[K:\mathbb{Q}]=2$ to knowing that $K = \mathbb{Q}[x]/a_2x^2 + a_1x + a_0$. ...
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1answer
373 views

For field extensions $F\subsetneq K \subset F(x)$, $x$ is algebraic over $K$

Let $x$ be an element not algebraic over $F$, and $K \subset F(x)$ a subfield that differs from $F$. Why is $x$ algebraic over $K$? Thanks a lot!
2
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1answer
119 views

The $p^{th}$ power of the elements of a basis of a finite separable field extension.

I came across the following claim. Let $L/K$ be a finite, separable extension of characteristic $p$ fields. Suppose $a_1,\dots,a_d$ is a basis. Then, so is $a_1^p,\dots,a_d^p$. To prove this, one ...
5
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1answer
148 views

Is $i\notin \mathbb{Q}(\zeta_p)$ for all odd primes $p$?

My main question is the title: for an odd prime $p$, denote a primitive $p^{\text{th}}$ root of unity by $\zeta_p$. Is it true that $i$ is not contained in the cyclotomic extension ...
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2answers
126 views

Unique isomorphism between fields generated by a domain.

Suppose $F$ and $K$ are fields both generated by a common subring $D$, which is a domain. My question is, why is there a unique isomorphism between $F$ and $K$ which is the identity on $D$? Wouldn't ...
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1answer
141 views

Finding the matrix of multiplication by $\theta^2$, where $\theta^3 - 3\theta + 1 = 0$

This is a problem from a on-line source which yet comes with a solution (self-studier; not h.w.). Let $E = \mathbb Q(\theta)$, where $\theta$ is a root of the irreducible polynomial \[ X^3 -3X + 1. ...
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1answer
151 views

Intersection of compositum of fields with another field 2

The following is a previous question with an additional hypothesis: Let $F_1$, $F_2$ and $K$ be fields of characteristic $0$ such that $F_1\cap K=F_2 \cap K=M$, the extensions $F_i/(F_i \cap K)$ are ...
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1answer
130 views

Intersection of compositum of fields with another field

Let $F_1$, $F_2$ and $K$ be fields of characteristic $0$ such that $F_1 \cap K = F_2 \cap K = M$, the extensions $F_i / (F_i \cap K)$ are Galois, and $[F_1 \cap F_2 : M ]$ is finite. Then is $[F_1 F_2 ...
5
votes
2answers
455 views

Primitive element of $\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q}$

Is there a clever way to determine a primitive element of the finite extension $$F=\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q} \text{ ?}$$ On simpler examples, I've been able to find one by ...
27
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2answers
2k views

Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
6
votes
1answer
149 views

An “independence” condition on two algebraic elements over $K$.

Let $K$ be a field and let $a,b\in \overline K$ be algebraic elements. I've stumbled upon a certain condition on $a,b$, which I feel could be considered an "independence" condition. I would like to ...
3
votes
2answers
193 views

Dedekind complete ⇒ Sequentially complete

Let F be an ordered field with least upper bound property. 1.Let $\alpha: \mathbb{N} \to F$ be a Cauchy sequence. Since F is an ordered field, $x$ is bounded both above and below. 2.By assumption and ...
2
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1answer
109 views

Homomorphism from $\mathbb{Q}$ to an ordered field F

I know that there exists a unique injective function $\gamma : \mathbb Q →F$ for any ordered field F. I don't understand why 'Prove $\gamma(r) = r•1_F$ for every $r\in \mathbb Q$' is an exercise.. ...
6
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1answer
737 views

Complete ordered field

I'm trying to prove that; If any Cauchy sequence is convergent in an ordered field F, every nonempty subset of F that has an upperbound has a sup in F. Let A be a nonempty subset of F that is not a ...