Question: Let K be an algebraically closed field. Show that every isomorphism of K onto a subfield of itself such that K is algebraic over

Let K be an algebraically closed field. Show that every isomorphism σ of K onto a subfield of itself such that K is algebraic over σ[K] is an automorphism of K, that is, is an onto map.

Step by Step Solution

★★★★★

3.47 Rating (170 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

Now K K is an isomorphism so 1 K K is an isomorphism Because K is algebraically closed and is algebraic ... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related A First Course In Abstract Algebra Questions!

Let k be an integer and set Compute (a) ||Ak|| (b) ||Ak||2 (c) p(Ak) (d) Explain why every Ak is a convergent matrix, even though their matrix norms can be arbitrarily large. (e) Why does this not...

Let k be an arbitrary number and -1 See Problem 29? k(k 1(k - 2). .(k n) n! lim

Let E be an algebraically closed extension field of a field F. Show that the algebraic closure F E of F in E is algebraically closed.

Let E be an algebraic extension of a field F. Show that every isomorphism of E onto a subfield of F leaving F fixed can be extended to an automorphism of F.

Mark each of the following true or false. ___ a. Let , E, where E F is a splitting field over F. Then there exists an automorphism of E leaving F fixed and mapping a onto if and only if irr(, F)...

We saw in Corollary 23 .17 that the cyclotomic polynomial s irreducible over Q for every prime p. Let be a zero of p (x), and consider the field Q(). a. Show that , 2 , , p-1 are distinct...

Mark each of the following true or false. ___ a. For all , E, there is always an automorphism of E mapping onto . ___ b. For , algebraic over a field F, there is always an isomorphism of F() onto...

Solve the following questions of probability and Algebra. 11:06 Lab manuals Putting it All Together Following the same format as the examples, write text descriptions for the algebra needed to arrive...

Let F be an algebraic closure of the field Q of rational numbers and let E F be a splitting field over Q of the set S = { x 2 + a | a Q} so that E is algebraic and Galois over Q (Theorem 3.11). (a)...

Let a be algebraic of degree n over F. Show from Corollary 48 .5 that there are at most n different isomorphisms of F() onto a subfield of F and leaving F fixed. Data from Corollary 48.5 Let be...

When the following compound is treated with sodium methoxide in methanol, two elimination products are possible. Explain why the deuterated product predominates by about a 7:1 ratio (refer to Problem...

Find the Fourier transform of f(x) Show details. f(x) = f(x) 2ix if -1

The Miscellaneous Revenue account should only be analyzed if it is material in amount. Group startsTrue or False

List five functions of exercise imagery.

The following scatterplot represents scores on Test 2 and Test 3 in Introductory Statistics for a random sample of students. The regression equation for these data is yÌ = 55.2 + 0.3708x,...

Reconsider the previous exercise. a. Determine the predicted score on Test 3 for a student who scores 55 on Test 2. b. Determine the predicted score on Test 3 for a student who scores 95 on Test 2....

Read FAQ 10.3.1. In your own words explain why using the sum of the absolute deviations is problematic in some scatterplots and, in turn, why the sum of squared deviations is preferred.

True or false 1. IRR is the expected return that is forgone by investing in a project rather than in comparable financial securities. _____2. Agency costs are incurred when the managers of a firm are...

Question 6 The book value of a firm's plant assets are: A . Generally greater than the market value when fixed assets are included. B . Based on historical cost. C . Equivalent to the firm's market...

Marathon Technologies, Inc. is using the modified internal rate of return (MIRR) when evaluating projects. The company is able to reinvest cash flows received from the project at an annual rate of...