Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Graduate chapters and video availability01 · Read and understand
What you will learn
- Identify automorphisms that fix the rational base field.
- Justify the conclusion "Gal(Q(√2)/Q)≅C₂" using the stated assumptions.
Before you start
Fields and irreducibility of x²−2 over Q.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
What are the Q-automorphisms of Q(√2)?
Why this math matters
Identify automorphisms that fix the rational base field. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Automorphisms fix every rational number.
- The extension is considered over Q, not over R.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Find the symmetries of a quadratic field extension
PausedQuestion: Start with the question. Paused.
Question
Start with the question
What are the Q-automorphisms of Q(√2)?
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
An automorphism sends √2 to a root of t²−2
It fixes rational coefficients and preserves polynomial equations.
Work through the mathematics
The possible images are √2 and −√2
Each choice extends to a+b√2 ↦ a±b√2.
Check the conclusion
Gal(Q(√2)/Q)≅C₂
The nonidentity conjugation squares to the identity, giving a two-element group.
The result
Gal(Q(√2)/Q)≅C₂
The nonidentity conjugation squares to the identity, giving a two-element group.
Common mistakes to catch
- An automorphism cannot send a root to an arbitrary real number.
- The base field determines which automorphisms are allowed.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is fixed by the nontrivial automorphism?
Show a hint
Solve a+b√2=a−b√2.
Reveal answer and explanation
Exactly Q
The equation forces b=0.
Practice 2
Compute the norm of 3+√2 to Q.
Show a hint
Multiply its two conjugates.
Reveal answer and explanation
7
(3+√2)(3−√2)=9−2.
Take the idea with you
Use conjugation to rationalize the reciprocal of a quadratic irrational.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: See a linear operator as a polynomial-module action
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