Suppose that F is a field of characteristic 0 and E is the splittingfield for some polynomial over F. If Gal(E/F) is isomorphic toZ20 ⊕ Z2, determine the number of subfields L of E there are suchthat L contains F anda. [L:F] = 4.b. [L:F] = 25.c. Gal(E/L) is isomorphic to Z5.

Answered: Suppose that F is a field of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

Suppose that F is a field of characteristic 0 and E is the splitting
field for some polynomial over F. If Gal(E/F) is isomorphic to
Z₂₀ ⊕ Z₂, determine the number of subfields L of E there are such
that L contains F and
a. [L:F] = 4.
b. [L:F] = 25.
c. Gal(E/L) is isomorphic to Z₅.

Expert Solution

Step 1

It is given that $F$ is a field of characteristic $0$ and $E$ is the splitting field for some polynomial over $F$ .

It is also given that $G a l (E / F)$ is isomorphic to $Z_{20} \oplus Z_{2}$ .

It is required to determine the number of sub-fields $L$ of $E$ there are such that $L$ contains $F$ and $[L : F] = 4$ .

The symmetry of $Z_{20} \oplus Z_{2}$ will be determined by the multiplication,

$20 \cdot 2 = 40$

So, the order of sub-fields will be,

$\frac{40}{4} = 10$

Now, it can be observed that multiplication of 10 within 40 will be the sub-fields. So, the sub-fields will be,

$10, 20, 30$

Hence, there will be three sub-fields.

Step 2

It is required to determine the number of sub-fields $L$ of $E$ there are such that $L$ contains $F$ and $[L : F] = 25$ .

The symmetry of $Z_{20} \oplus Z_{2}$ will be determined by the multiplication,

$20 \cdot 2 = 40$

So, the order of sub-fields will be,

$\frac{40}{25} = 1.6$

It can be observed that this a fraction. So, no sub-field will be present.

Step by step

Solved in 3 steps

Knowledge Booster

$Ring$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions