Question

A be real positive 4th root of 2 find intermediate field extension

Answer 1

Answer:

Hope this helps you.

Presumably we're talking about extending the field of rationals.

K = field of rationals

M = K( α ) the simple extension of K obtained by adjoining α = 2^(1/4), the real positive 4th root of 2

We want a field L such that

K < L < M

Take L = K( √2 ), the simple extension of K obtained by adjoining the (positive) square root of 2.

As α² = √2, we have L ≤ M.

As the minimum polynomials of α and √2 over K are x⁴ - 2 and x² - 2, the extensions L : K and M : K have different degrees, so L ≠ M.

Also, √2 is not in K, so L ≠ K.

So

K < L < M

and L is an "intermediate" field extension.