- #1

kmitza

- 17

- 4

- TL;DR Summary
- I have run into the following problem, I have managed to solve some of it but I don't have idea for the rest.

We have Galois extension ## K \subset L ## and element ##\alpha \in L## and define polynomial $$f = \prod_{\sigma \in Gal(L/K)} (x - \sigma(\alpha))$$

Now we want to show that ## f \in K[x] ## which is relatively easy to see because we can take ##\phi(f)## for any ## \phi \in Gal(L/K) ## then ## \phi \circ \sigma ## ranges over the galois group and all the roots stay fixed and we're done.

Further we want want to prove that ## f ## is power of minimal polinomial of ## \alpha ## and that it is equal to the minimal polynomial iff ## L = K(\alpha) ##.

Any hints or help will be appreciated

Now we want to show that ## f \in K[x] ## which is relatively easy to see because we can take ##\phi(f)## for any ## \phi \in Gal(L/K) ## then ## \phi \circ \sigma ## ranges over the galois group and all the roots stay fixed and we're done.

Further we want want to prove that ## f ## is power of minimal polinomial of ## \alpha ## and that it is equal to the minimal polynomial iff ## L = K(\alpha) ##.

Any hints or help will be appreciated