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
36 views

Question about the quotient of two lattices.

Let $F$ be a non-archimedian local field with valuation $\nu$. Then $\mathcal{O}=\{x\in F: \nu(x)\geq 0\}$ is the ring of integers of $F$. $\mathfrak{m}=\{x\in F: \nu(x)> 0\}$ is the maximal ideal ...
2
votes
1answer
230 views

Intermediate fields of a field extension

Let $L := \Bbb Q(\sqrt 5,\sqrt 7).$ We have proved that $L/\Bbb Q$ is galois. I have to find all the intermediate fields of $L/\Bbb Q$. So far, I have found $\Bbb Q(\sqrt 5)$, $\Bbb Q(\sqrt 7)$, ...
1
vote
1answer
89 views

Question on the proof of existence of splitting fields for a family of polynomials

I have a question regarding the following well known result: Let $C\subseteq K[x]$ be a family of polynomials. We know that $C$ possesses a splitting field over $K$. The proof I am reading goes like ...
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0answers
55 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 ...
2
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0answers
73 views

Factor polynomials into irreducibles over GF(q)

The polynomials $x^4+x^3+1$ and $x^4+x^3+x^2+x+1$ are irreducibles over GF(2). (a) Factor both polynomials into irreducibles over GF(4). (b) Factor both polynomials into irreducibles over GF(8). I ...
2
votes
1answer
69 views

A field with irreducible polynomial that has multiple roots

Can you give me an example of a field $\mathbb{K}$ such that there exists a polynomial $p(x)\in\mathbb{K}[x]$ that is irreducible and has a multiple root?
2
votes
1answer
64 views

Problem concerning formally real fields

I'm trying to reconcile a fact I am reading in David Marker's Model Theory text. He claims on page 326 that $\mathbb{F}=\mathbb{Q}(\sqrt{2}, \sqrt{-2})$ is a formally real field. This seems like it ...
2
votes
4answers
91 views

Let $S=\big\{\sqrt[n]{3}\colon n\in \mathbb{N}\big\}$. Is the extension $\mathbb{Q}[S]\colon\mathbb{Q}$ algebraic?

A field extension $L\colon K$ is algebraic if every element in $\alpha \in L$ is algebraic over $K$. An elemenet $\alpha \in L$ is algebraic over $K$ if there exists a polynomial $f \in K[x]$ such ...
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vote
1answer
41 views

Find a polynomial whose splitting field is $\mathbb{Q}[\alpha,i]$

Let $f(x)=x^{3}-3x+1$ and let $\alpha$ be a root in $f$. i) Show that the polynomial $f$ is irreducible in $\mathbb{Q}[x]$. ii) Show $\alpha^{2}-2$ is a root of $f$ as well, and show that all roots ...
0
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0answers
39 views

Splitting field condition for subset of isomorphism image

Let $F$ be a field, and $K$ a finite extension of $F$. Suppose that for every isomorphism $h$ which domain $K$ that fixes $F$, we have $h(K)\subseteq K$. How can we show that $K$ is a splitting field ...
1
vote
1answer
112 views

Splitting field condition for all roots of irreducible polynomial

Let $F$ be a field, and $K$ a finite extension of $F$. Suppose that for every irreducible polynomial $P(x)\in F[x]$, if $P(x)$ has one root in $K$, then $P(x)$ has all its roots in $K$. How can we ...
0
votes
1answer
115 views

Injective homomorphism from extension field to complex numbers

Let $P(x)$ be irreducible in $F[x]$, where $F$ is a subfield of $\mathbb{C}$. Let $c$ be a complex root of $P(x)$. Let $h:F\rightarrow\mathbb{C}$ be an injective homomorphism. If $\deg P(x)=n$, I ...
1
vote
2answers
309 views

Splitting field of $x^n-a$ contains all $n$ roots of unity

This statement is suggested as a correction to this question: If $K$ is the splitting field of the polynomial $P(x)=x^n-a$ over $\mathbb{Q}$, prove that $K$ contains all the $n$th roots of unity. ...
2
votes
1answer
57 views

Calculating polynomials in a Galois field

I'm in $\text{GF}(8) = \text{GF}(2^3)$ and have an irreducbile polynomial $p(x) = x^3 + x + 1$, then $\text{GF}(8) = \mathbb{Z}_2[x]/\langle p(x) \rangle$ . Now I want to multiply $2$ elements of ...
1
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0answers
41 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) ...
0
votes
1answer
53 views

element algebraic over a field

How can I show that if $a,b$ are elements of a field $K$ and $k, l$ are element of the positive natural numbers, then the element $\sqrt[k]{a}\cdot\sqrt[l]{b}$ is also algebraic over $K$. I'm stuck ...
3
votes
0answers
66 views

Splitting field containing $n$th root

Let $K$ be a splitting field of a polynomial over $\mathbb{Q}$. Suppose $K$ contains an $n$th root of some number $a$. Then how can we show that $K$ contains all the $n$th roots of unity? I don't ...
0
votes
1answer
70 views

Splitting field is a splitting field for any root of irreducible polynomial

Suppose $K$ is a splitting field over $F$ such that $[K:F]=n$. Prove that $K$ is a splitting field over $F$ for any irreducible polynomial of degree $n$ of $F(x)$ having a root in $K$. Well, let ...
2
votes
0answers
56 views

Field extension of complex root of cubic equation

If $c$ is a complex root of a cubic $a(x)\in\mathbb{Q}[x]$, show that $\mathbb{Q}(c)$ is the splitting field of $a(x)$ over $\mathbb{Q}$. For this, we must show that $\mathbb{Q}(c)$ contains all ...
1
vote
1answer
47 views

Root field of $x^3+x^2+x+2$ over $\mathbb{Z}_3$

Find the root field of $p(x)=x^3+x^2+x+2$ over $\mathbb{Z}_3$. $p(x)$ is irreducible in $\mathbb{Z}_3$ by direct substitution of $x=0,1,2$. Suppose $u$ is a root of $p(x)$. Then ...
2
votes
0answers
62 views

Relationship between the minimal polynomial of $\sin{2^{\circ}}$ and $\sin{5^{\circ}}$ over $\mathbb Q$

Let $f(x)$ be the minimal polynomial of $\sin{2^{\circ}}$ over $\mathbb Q$, and $g(x)$ be the minimal polynomial of $\sin{5^{\circ}}$ over $\mathbb Q$, then $f(x)+f(-x)=2 g(x)\tag 1$. I find this ...
1
vote
1answer
87 views

Root field of $x^3+2x+1$ over $\mathbb{Z}_3$

Find the root field of $a(x)=x^3+2x+1$ over $\mathbb{Z}_3$. Suppose $u$ is a root of $x^3+2x+1$. Then $a(u+1)=(u+1)^3+2(u+1)+1=u^3+2u+1=0$, so $u+1$ is also a root, and similarly $u+2$ is also a ...
5
votes
3answers
138 views

What is $[\mathbb{Q}(i,\sqrt{2},\sqrt{3}):\mathbb{Q}]$?

What is $[\mathbb{Q}(i,\sqrt{2},\sqrt{3}):\mathbb{Q}]$? On the one hand, we have ...
1
vote
1answer
45 views

Element algebraic over field iff algebraic over field extension

Let $F$ be a field, and $K$ a finite extension of $F$. I want to show that any element algebraic over $K$ is algebraic over $F$, and conversely. Well, if $a$ is algebraic over $F$, we can write ...
0
votes
2answers
69 views

Coprime degree of minimum polynomials

Let $F,K$ be fields. Suppose $a,b\in K$ are algebraic over $F$ with degrees $m,n$, where $(m,n)=1$. I want to show that $F(a,b)$ is of degree $mn$ over $F$, and $F(a)\cap F(b)=F$. Consider ...
2
votes
1answer
27 views

Extension elements always exist for field extensions?

I'm trying to understand the concept of field extensions. If $A\subseteq B$ are fields (i.e. $B$ is an extension of $A$), then we can view $B$ as a vector space over $A$ (i.e. $A$ is the field of ...
0
votes
1answer
53 views

No field between extension fields with prime degree

Let $F$ be a field, and $K$ a field extension of $F$. Prove that if $[K:F]$ is prime, then there is no field $L$ such that $F\subset L\subset K$ and $F\neq L,L\neq K$. Well, if $L$ is an extension of ...
4
votes
2answers
100 views

Field containing sum of square roots also contains individual square roots

Let $F$ be a field of characteristic $\neq 2$. Let $a\neq b $ be in $F$. Suppose $\sqrt{a}+\sqrt{b}\in F$. Prove that $\sqrt{a}\in F$. We have $(\sqrt{a}+\sqrt{b})^2=a+b+2\sqrt{ab}\in F$, so ...
4
votes
2answers
367 views

Find basis of $\mathbb{Q}(\sqrt{2}+\sqrt[3]{4})$

Find a basis of $\mathbb{Q}(\sqrt{2}+\sqrt[3]{4})$ over $\mathbb{Q}$. I think the basis should be $1, \sqrt{2}, \sqrt[3]{4}, \sqrt[3]{4}\cdot\sqrt{2},\sqrt[3]{4}^2,\sqrt[3]{4}^2\cdot\sqrt{2}$. So I ...
0
votes
2answers
94 views

Element in field of quotients is transcendental

Let $F\subseteq E$ be fields, and let $c\in E$. Let $F(c)$ be the field of quotients containing $F$ and $c$. Suppose $c$ is transcendental over $F$. Prove that every element in $F(c)$ but not in ...
1
vote
2answers
188 views

Quadratic extension is generated by square root in field

Let $F$ be a field whose characteristic is $\neq 2$. Suppose the minimum polynomial of $a$ over $F$ has degree $2$. Prove that $F(a)$ is of the form $F(\sqrt{b})$ for some $b\in F$. Well, $F(a)$ ...
2
votes
4answers
55 views

Field $F[x]/\langle p(x)\rangle$ contains both roots of $p(x)$ if degree $2$

Consider this question If $p(x)$ is irreducible and has degree $2$, prove that $F[x]/\langle p(x)\rangle$ contains both roots of $p(x)$. I'm wondering if it's poorly phrased? The field ...
0
votes
2answers
158 views

The number of possible extensions of an embedding of a field into a algebraically closed field.

I've been thinking this more than a week. My problem is the second part of the Propositon 2.7. in Algebra by Serge Lang. In the proposition, k is a field and $\sigma:k \rightarrow L$ is an embedding ...
2
votes
1answer
79 views

Field generated by element contains all other roots of minimum polynomial?

I just want to check my understanding of this point of field theory: Let $F$ be a field, and $c$ algebraic over $F$ with minimum polynomial $p(x)$. Then we know that $F(c)$ (the field generated by ...
1
vote
1answer
72 views

Set of Functions is a Vector Space problem

Let $F$ be a field. Consider the $F$-vector space $F(X, F)$ of all functions from $X = \{ 1, 2 \} \to F$. Define $e_1, e_2 \in F(X, F)$ by $e_1 = \{ (1, 1),(2, 0) \}$ and $e_2 = \{ (1, 0),(2, 1) \}$. ...
1
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2answers
146 views

Applications of additive version of Hilbert's theorem 90

Additive version of Hilbert's theorem 90 says that whenever $k \subset F$ is cyclic Galois extension with Galois group generated by $g$, and $a$ is element of $L$ with trace 0, there exists an ...
4
votes
1answer
97 views

$\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ ??

Let $p,q$ be primes, $p≠q$, then I have to show that $\mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q})$ So far I've tried a lot of things with minimal polynomials and bases, ...
2
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1answer
71 views

Understanding a Proof in Galois Theory

The following is an extract from my Galois Theory course lecture notes. I understand the proof in the reverse direction so have included only the part of the proof that confuses me, even though it ...
2
votes
3answers
130 views

How to prove the one-variable calculus definition of derivative extends to $\Bbb C$ *only* because $\Bbb C$ is a field?

I have been told the one-variable calculus definition of derivative extends to $\Bbb C$ only because $\Bbb C$ is a field. See : Higher dimensional analogues of the argument principle? $$ ...
3
votes
7answers
378 views

What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring?

The title pretty much says it all. Of course, one answer (IMO unsatisfactory) to such questions goes something like "a definition is a definition, period." In my experience, mathematical definitions ...
0
votes
3answers
109 views

Let $F/K$ be a field extension and assume that $[F:K]=3$. Can we conclude that $F=K(\sqrt[3]{a})$ for all $a \in K$? [closed]

Let $F/K$ be a field extension and suppose that $[F:K]=3$. Can we conclude that $F=K(\sqrt[3]{a})$ for all $a \in K$? Thanks a lot.
9
votes
1answer
128 views

Does there exist a field $(F,+,*)$ so that $(F,+) \cong (F^*,*)$?

This question occurred to me earlier today. I can see that if the field has a unit, then there is an element of multiplicative order $2$, namely $-1$. Thus if there was an isomorphism $(F,+) \cong ...
0
votes
0answers
29 views

Prove that if $f$ is a primitive polynomial over $F_q$ then $f$ divides $Q_{q^m-1}$.

I am not writing my complete proof, and my conclusion is that since all the roots of $f$ are primitive $(q^m-1)st$ roots of unity and so are the roots of $Q_{q^m-1}$. Therefore, $f$ must divide ...
24
votes
1answer
346 views

Rigidity of the category of fields

Let's call a category rigid if every self-equivalence is isomorphic to the identity. For example, $\mathsf{Set}$, $\mathsf{Grp}$, $\mathsf{Ab}$, $\mathsf{CRing}$ (MO/106838), $\mathsf{Top}$ ...
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0answers
40 views

Properties of algebraic closure of finite field

I want to ask if these statements are true, and can anyone please give me some reference/proof if possible: Suppose k is a finite field with algebraic closure $\bar{k}$. 1) Do we have $\bar{k}^*$ ...
0
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1answer
122 views

Locally finite field

Definition: A field is locally finite if every its finitely generated subfield is finite. Show that a field $K$ is locally finite iff it is embeddable to the algebraic closure of $F_p$, for some ...
2
votes
1answer
120 views

Fields and proper subfields. [duplicate]

Specific question: Let $F$ be a field and assume that $\mathbb{Q}$ is a proper subfield of $F$. Can $F$ be isomorphic to $\mathbb{Q}$? Studying the foundaments of field theory I have to ask: Can ...
11
votes
2answers
307 views

Can A Decidable Theory Have Nonrecursive Models?

Tennenbaum's theorem proves neither addition nor multiplication can be recursive in any countable non-standard model of arithmetic. Tennenbaum's proof applies to theories much weaker than PA. ...
1
vote
1answer
120 views

Proving that set is (or is not) a field

Let $P = \{a + b\sqrt[3]3 + c\sqrt[3]9, a, b, c \in \Bbb Z \}$ It is easy to prove that $(P, +, \cdot)$ is a ring considering ordinary addition and multiplication. How to prove that this set is or is ...
0
votes
1answer
51 views

book on cubic fields by Kisilevsky

I have been trying to get a copy of Indices in cyclic cubic fields, in “Number Theory and Algebra”, Academic Press, 1977 by D. S. Dummit and H. Kisilevsky but it is nowhere to be found. Does anyone ...