Quadratic forms with p-adic coefficient

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SUMMARY

The discussion focuses on determining the p-adic fields in which the quadratic form 3x² + 7y² - 15z² represents zero. Key calculations involve the Hasse Invariant and the discriminant of the form. The condition for representation is established through the theorem stating that the Hasse Invariant must equal (-1, -d(f)), where d(f) is the discriminant and (,) denotes the Hilbert Symbol.

PREREQUISITES
  • Understanding of quadratic forms
  • Knowledge of p-adic fields
  • Familiarity with Hasse Invariant
  • Concept of discriminants in algebra
NEXT STEPS
  • Study the calculation of Hasse Invariant in detail
  • Research the properties of Hilbert Symbols
  • Explore theorems related to quadratic forms and their representations
  • Learn about discriminants and their role in algebraic structures
USEFUL FOR

Mathematicians, algebraists, and students studying number theory or quadratic forms, particularly those interested in p-adic analysis.

antonio85
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How can I found out in which p-adic fields a quadratic form represent 0?

For example in which p-adic fields does the form 3x2+7y2-15z2 represent zero?
 
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You need to calculate Hasse Invariant and the discriminant of this form.

After that there's a theorem that gives conditions for which this form represents zero.
I can't recall the condition but I think Hasse Invariant, should be equal
(-1,-d(f)) where d(f) is the discriminant and (,) is Hilbert Symbol, after now after I checked my notebook, yes I am correct.

****, what a memory...
 

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