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
193 views

Notation for finite fields

What is the meaning of the following: If $q(x)$ is irreducible polynomial of degree $d$ and $d$ divides $n$, then $q(x)$ divides $x^{p^n}-x$. Let $F = F_p[x]/(q(x)) = F_p[\alpha]$ where ...
3
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1answer
332 views

Non-Archimedean Fields

Are there non-Archimedean fields without associated valuation or being a non-archimedan field implies it is a valuation field? I understand that a non-Archimedean field is a field which does not ...
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2answers
260 views

How to determine if this is a field?

A finite subring $R$ of a field $V$ contains $1$ (so $1$ is an element of $R$). The question is: True or False: The ring $R$ must be a field. I thought that if $R$ was a field it had to be a finite ...
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1answer
223 views

Primitive Element for Field Extension of Rational Functions over Symmetric Rational Functions

A rational function $f$ in $n$ variables is a ratio of $2$ polynomials, $$f(x_1,...x_n) = \frac{p(x_1,...x_n)}{q(x_1,...x_n)}$$ where $q$ is not identically $0$. The function is called symmetric if ...
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5answers
2k views

Can you construct a field with 6 elements? [duplicate]

Possible Duplicate: Is there anything like GF(6)? Could someone tell me if you can build a field with 6 elements.
3
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1answer
194 views

Commutative Algebra - Polynomial Rings

Let $Z$ be the ring of integers, $p$ a prime and $F_p = Z/pZ$ the field with $p$ elements. Let $x$ be an indeterminate. Set $R_1 = F_p[x]/(x^2-2)$, $R_2 = F_p[x]/(x^2-3)$. Determine whether the rings ...
7
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2answers
271 views

Splitting field of $x^{13}+1$ over $\mathbb{Q}$

I want to know how to calculate the degree of the field extension $[K:Q]$ where $K$ is the splitting field of $x^{13}+1$ over $\mathbb{Q}$. I'm new to this area and this is not really covered in my ...
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1answer
64 views

Extending $\phi: A \rightarrow \Omega$ to $A[x] \rightarrow \Omega$ where $A$ is integral domain and $x$ transcendental over $A$

Let $A \subseteq B$ be integral domains and let $\phi:A \rightarrow \Omega$ be a homomorphism of $A$ into the infinite algebraically closed field $\Omega$. Let $x \in B$ and suppose that $x$ is ...
5
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1answer
175 views

Finding the splitting field of $f(x)$

I'm trying to learn the theory of splitting fields. So I went through this example on an old exam: Find the splitting field $K$ of $f(x)$ over $\mathbb{Q}$ for $f(x)=x^6-9$ $x^6-9=(x^3-3)(x^3+3)$ and ...
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1answer
219 views

Generator of the multiplicative groups of units in $\mathbb{Z_5[x]}/(x^{2}+3x+3)$

I would like to find a generator of the multiplicative groups of units in $\mathbb{Z_5[x]}/(x^{2}+3x+3)$. This is a field since $x^{2}+3x+3$ is irreducible, so every coset with $bx+a\not=0$ as a ...
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1answer
273 views

Extension of Homomorphisms (Lang, Atiyah and McDonald)

Let $A$ be a subring of a field $K$, and suppose that $A$ is a local ring with maximal ideal $\mathfrak{m}$. Let $x \in K, \, x \neq 0$. Let $\phi: A \rightarrow L$ be a homomorphism of $A$ into the ...
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2answers
104 views

Extension of an embedding to a field extension by a transcendental element

Let $K$ be a subfield of a field $K'$ and suppose we have an embedding $\phi:K \rightarrow L$ of $K$ into an algebraically close field $L$. Let $x \in K'$. If $x$ is algebraic over $K$, then we can ...
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5answers
241 views

Find an ideal $I$ of $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/{I}$ is a field

Find an ideal $I$ of $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/{I}$ is a field. How can one justify the answer in the shortest number of lines?
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1answer
77 views

Automorphism and Stabilizer

Let $K \leq E \leq F$ be fields such that $[E:K]=n$. Let $Aut_K F$ act on the set $S$ of intermediate fields where $\sigma(I)$ gives the action of $\sigma \in Aut_K F$ on an intermediate field $I$. ...
3
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4answers
191 views

Nonzero elements of splitting field

Let $F$ be a splitting field of $x^{p^{n}} - x \in \mathbb{Z}_p[x]$. How is it that the nonzero elements multiply to $-1$ and sum to $0$? I don't get how we get that result.
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What kind of work do modern day algebraists do?

Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
3
votes
1answer
111 views

Trace and Norm maps on differential extensions

I'm working through a proof which is rather algebraic, and my abstract algebra is probably only basic to intermediate. I have a differential extension $E/K$ of a differential field $K$, and the proof ...
2
votes
1answer
96 views

Latin squares of even order with all cells only participating in one subsquare.

For even ordered Latin squares, we can create squares in which every cell participates in a $2\times 2$ subsquare by using a simple circulant as for $n=6$ below: ...
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vote
2answers
202 views

Irreducible polynomials with integer coefficients over Q

Suppose p(x) is an irreducible polynomial over Q of degree n, with integer coefficients. If p(x) has two roots r1 and r2 satisfying r1r2 = 5, prove that n is even. Attempt at solution: Because the ...
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1answer
93 views

Proof about Field conjugation isomorphisms

I'm having an awful time making sense of a proof and I was hoping someone could help. Theorem: Let $\alpha$ and $\beta$ be algebraic over a field $F$ with $deg(\alpha, F) = n$, as elements of a ...
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1answer
71 views

Maximal Separable Subextension is Finite?

Consider the following statement: "Let $L/K$ be an algebraic field extension. Then the maximal separable sub-extension is finite." Here is what seems to be a proof: "Let $M/K$, $K \subset M \subset ...
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2answers
393 views

Number of field homomorphisms from an extension field of $\mathbb Q$ to $\mathbb C$

Take $\mathbb{Q}$ $\subset$ $K$ $\subset$ $\mathbb{C}$ with $[K:\mathbb{Q}]$ finite. How would you show that the number of field homomorphisms from $K$ to $\mathbb{C}$ is equal to $[K: ...
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2answers
876 views

How can one express $\sqrt{2+\sqrt{2}}$ without using the square root of a square root?

I was trying to review some analysis, and came across problem 3 from page 78 of Walter Rudin's Principles of Mathematical Analysis. As part of the problem, I wanted to try to write ...
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1answer
142 views

Relation of compositum of fields

Let $E/k$ be a finite field extension, $\operatorname{char}(k)=p>0$. Suppose that $E^p k = E$. Is it then true that $E^{p^n}k = E$ for any positive integer $n$? If yes, why? Thanks.
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3answers
108 views

Uniqueness of prime-power fields

I'm still stuck on the proof of the following theorem. I've asked two questions so far to get to where I am even at this point. Theorem: Let $p$ be a prime and let $n\in\mathbb{Z}^{+}$. If $E$ and ...
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3answers
97 views

Question about algebraic field extensions

If I have a subfield $F$ of a field $E$, and an algebraic (over $F$) $\alpha\in E$, I can form $F(\alpha)$ which is isomorphic to $F[x]/\langle f(x)\rangle$ for $f(x) = irr(\alpha, F)$. That is, ...
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1answer
82 views

Can't follow a proof involving Prime-Power Fields

Theorem: Let $p$ be a prime and let $n\in\mathbb{Z}^{+}$. If $E$ and $E'$ are fields of order $p^{n}$, then $E\cong E'$. Proof: Both $E$ And $E'$ have $\mathbb{Z}_{p}$ as prime fields (up to ...
3
votes
1answer
119 views

Is an automorphism of a normal extension determined by its image of the maximal separable sub extension?

Let $L / K$ be a normal, algebraic field extension. Suppose that the maximal separable sub- extension $M/K$ is finite, $K \subseteq M \subseteq L$. By the primitive element theorem, $M=K(x)$ for some ...
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2answers
111 views

Question about isomorphism between a ideal and a polynomial ring

Sorry for my ignorance, my question is: Let be $F[X]$ a polynomial quotient ring, where $F$ is a finite field with characteristic 2. Are there any ideal, $I$, such that $I$ is isomorphic to $F[X]$?.
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1answer
90 views

Question about a corollary about Finite Fields

Definition: A field extension $E$ of $F$ is of degree $n$ (and is called a finite field extension) if $E$ is an $n$-dimensional vector space over $F$. Theorem: Let $E$ be a degree $n$ finite ...
4
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1answer
119 views

Unramified extension is normal if it has normal residue class extension

Let $K/F$ be an unramified extension such that $\rho_K / \rho_F$ (the corresponding extension of residue classes) is normal. Prove $K/F$ is normal. I guess I need to do some polynomial lifting, but ...
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1answer
149 views

What is a valuation associated to an ordering on a field?

If $(K,\leq)$ is a totally ordered field with $P\!=\!\{\alpha\!\in\!K;\, 0\!\leq\!\alpha\}$, how is the valuation associated to $P$ defined? I was searching through Prestel & Delzell's Positive ...
3
votes
1answer
292 views

A formula for the roots of a solvable polynomial

Let $F$ be a field and $p(x)\in F[x]$ a separable polynomial, denote $K$ as the splitting field of $p$ and assume that $K/F$ is Galois with a solvable Galois group. I don't understand if this imply ...
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votes
2answers
149 views

Is a polynomial solvable by roots iff every irreducible factor is?

Let $F$ be a field, I asked myself if $p(x)\in F[x]$ is solvable by radicals iff every irreducible factor is solvable by radicals. My thoughts: If every irreducible factor is solvable by roots then ...
2
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1answer
265 views

Irreducible Polynomials in Finite Fields

I'm reading through some notes online concerning finite fields, and attempting to come up with a proof that all finite fields of the same size are isomorphic. But I'm getting stuck at a certain point, ...
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1answer
51 views

Basic question about fractions

I'm solving some exercises about fields and am trying to find the inverse for $a_1 + \sqrt{2}b_1$, i.e. $\frac{1}{a_1 + \sqrt{2}b_1}$. This means I need to split the fraction into something of the ...
4
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4answers
163 views

How many elements in the finite field $F_{256}$ satisfy $x^{103}=x$?

How many elements of the finite field $\mathbb{F}_{256}$ with 256 elements satisfy $x^{103}=x$?
3
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1answer
286 views

The order of the Galois group of a cyclotomic field over a finite prime field [duplicate]

Possible Duplicate: For what $(n,k)$ there exists a polynomial $p(x) \in F_2\[x\]$ s.t. $\deg(p)=k$ and $p$ divides $x^n-1$? Galoisgroup $\operatorname{Gal}(K(\mu_n) / K) \subseteq (\mathbb{Z} ...
0
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1answer
774 views

Cyclotomic polynomial over a finite prime field [duplicate]

Possible Duplicate: Irreducible factors of $X^p-1$ in $(\mathbb{Z}/q \mathbb{Z})\[X\]$ Let $p$ be a prime number. Let $l$ be an odd prime number such that $l \neq p$. Let $X^l - 1 \in ...
3
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1answer
514 views

Subgroup of Galois group of polynomial over $\mathbb{Q}$

Let $K$ be the splitting field of $x^5-3 \in \mathbb{Q}[x]$. We can see $K = \mathbb{Q}(3^{1/5}, \zeta_5)$ where $\zeta_5 = e^{2 \pi i/5}$, and $[K: \mathbb{Q}] = 20$. It's easy to see $\sqrt{5} \in ...
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2answers
313 views

Proving there are no subfields

I am trying to solve Q11 at pg. 582 from the book Abstract algebra by Dummit and Foote, the question is: Let $f\in\mathbb{Z}[x]$ be an irreducible quartic whose splitting field has Galois group ...
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1answer
430 views

Positivity of the norm of an element of a cyclotomic field

Let $l$ be an odd prime number and $\zeta$ be an $l$-th primitive root of unity in $\mathbb{C}$. Let $\mathbb{Q}(\zeta)$ be the cyclotomic field and $\alpha$ be a non-zero element of ...
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1answer
91 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 ...
1
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2answers
4k 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
218 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
394 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
463 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
63 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 ...
1
vote
1answer
58 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 $g(x^{p})=(g(x))^{p}\mod p$ ...