0
votes
0answers
33 views

Extending a morphism to a finite algebraic field extension [duplicate]

I am trying to understand the proof of Theorem 5.21 in Introduction to Commutative Algebra, and am stuck on the portion underlined in red (note that $$\Sigma := \{(A,f) \mid A \text{ is a subring of ...
5
votes
1answer
76 views

$Gal(\mathbb{Q}(\sqrt 2 + \sqrt 3)/\mathbb{Q})$

A basis for $\mathbb{Q}(\sqrt 2 + \sqrt 3)$ over $\mathbb{Q}$ is $\{1,\sqrt 2 , \sqrt 3 , \sqrt 6 \}$ The roots of $x^2 -2$ are $\pm \sqrt 2$ and the roots of $x^2 -3$ are $\pm \sqrt 3$ so to find ...
3
votes
1answer
79 views

Morphisms, Splitting Fields, and Primitive Cube Roots

Let $\alpha=\sqrt[3]2$ and $\omega=e^{2\pi i/3}$, and let $K=\mathbb{Q}[\alpha,\omega]$ be the splitting field of $f(x)=x^3-2$ over $\mathbb{Q}.$ Determine all morphisms $K\to K$. Ok, so I have ...
2
votes
2answers
71 views

Proving the Frobenius map is an endomorphism

I have prime $p$, and $K$ a field such that $p \cdot 1 = 1+1+\cdots+1 = 0$. I am asked to prove that $F: K \rightarrow K$, $a \mapsto a^p$ is a ring homomorphism. I can prove this for ...
0
votes
1answer
44 views

Say $K=\mathbb Q(2^{1/3})$. Determine all endomorphisms of $K$.

Say $K=\mathbb Q(2^{1/3})$. Determine all endomorphisms of $K$, and justify your answer. Hint: Say $f(x)= x^3-2$. How many roots of $f$ are in $K$? For this I know $x^3-2$ has 1 real root, ...
1
vote
2answers
172 views

Is the ring $\mathbb{Z}_5[x]$ isomorphic to the ring of polynomial functions from $\mathbb{Z}_5$ to $\mathbb{Z}_5$?

Is the ring $\mathbb{Z}_5[x]$ isomorphic to the ring of polynomial functions from $\mathbb{Z}_5$ to $\mathbb{Z}_5$? If not, what is a good counterexample? If yes, how can we prove that there's a ...
5
votes
1answer
95 views

A question on morphisms of fields

Let $A,B$ be two fields. Let $\phi:A\rightarrow B$ and $\psi:B\rightarrow A$ be two morphisms of fields. Can i conclude that $A$ and $B$ are isomorphic fields? My guess is yes, because every morphism ...
0
votes
1answer
51 views

Extensions of $\sigma:\mathbb{Q}(\xi)\rightarrow\mathbb{Q}(\xi), \ \sigma(\xi)=\xi^{2}$ to $\mathbb{Q}(\xi)(\sqrt[3]{2})$

Let $L$ be the splitting field of $T^{3}-2$ and let $\sigma:\mathbb{Q}\left(\xi\right)\rightarrow\mathbb{Q}\left(\xi\right)$ be a morphism defined by $\sigma\left(\xi\right)=\xi^{2}$. Find all ...
0
votes
1answer
62 views

Extensions of $\mathbb{Q}\left(\sqrt{2}\right)\rightarrow\mathbb{Q}\left(\sqrt{2}\right),\ \sqrt{2}\mapsto -\sqrt{2}.$

I have to solve an exercise that requires to find all extensions $\hat{\sigma}:\mathbb{Q}\left(\sqrt{2},\xi\right)\rightarrow\mathbb{Q}\left(\sqrt{2},\xi\right)$ (with $\xi$ being a third root of the ...